UC-NRLF 


IN  MEMORIAM 
FLOR1AN  CAJORI 


" 


^n 


OUTLINES 


-OF- 


NUMBER  SCIENCE 


-BY- 


NATHAN  NEWBY, 

Professor  of  Mathematics  in  the 

INDIANS  STATE  NORMAL  SCHOOL, 


TERRE  HAUTE,  INDIANA. 


SECOND  EDITION. 


TERRE  HAUTE : 

GEORGE  H.  HEBB  PRESS. 

1884. 


[Entered  according  to  Act  of  Congress  in  the  year  1883. 

by  NATHAN  NEWBY, 
in  the  office  of  the  Librarian  of  Congress.] 


^Jd 


EXPlLANATORY. 

This  little  book  is  prepared  to  meet  the  author's 
convenience  in  giving  Arithmetical  instruction  in  the 
Normal  School,  and  at  the  solicitation  of  a  large  num- 
ber of  pupil-teachers,  who  have  received  in  the  class- 
room and  have  afterwards  used  in  their  own  schools, 
much  of  the  matter  herein  contained.  The  work  is 
but  an  elaboration  of  a  series  of  articles  written  by  the 
same  hand  in  1873,  and  published  in  an  educational 
paper  in  this  State.  An  examination  of  the  order  of 
procedure  will  show  that  the  book  is  not  made  for 
children,  but  for  the  mature  mind  already  conversant 
with  most  of  the  facts  of  number  as  presented  in  the 
text  books  on  Arithmetic.  "  The  aim  of  the  Normal 
School  is  not  so  much  to  teach  the  facts  of  the  common 
school  branches  as  to  make  a  thorough  study  of  the  re- 
lations of  those  facts  to  one  another.  The  study  of 
these  relations  opens  up  new  lines  of  thought  that 
mate  the  common-school  branches  intensely  interest- 
ing studies  to  most  students." 

The  province  of  the  school  as  thus  stated,  being 
kept  in  view,  the  attempt  has  been  made  so  to  present 
the  topics  of  number  science  that  the  lines  which  unify 
parts  that  are  kindred,  may  be  readily  seen. 

The  discussion  begins  with  a  general  definition  of 
M  athematics,  together  with  a  brief  consideration  of  the 


realms  of  Space  and  Time,  These  are  defined  as  condi- 
tions:— Space  as  the  condition  of  extension,  and  hence 
preliminary  to  Geometry  in  its  various  phases;  and 
Time  as  the  condition  of  succession,  and  hence  prelim- 
inary to  the  idea  of  Number. 

Starting  with  Time  as  the  conditioning  factor,  it 
is  sought  to  show  the  genesis  of  number  in  general. 
From  this  idea  the  mind  readily  passes  to  that  of  num- 
ber in  particular.  The  term  Integral  unit  or  Unit  one 
(from  Davies),  is  fixed  upon  to  designate  the  primary 
idea  of  number  in  particular.  It  must  be  borne  in  mind 
that  the  elements  which  enter  into  a  science  are  men- 
tal and  not  material  objects.  A  pencil,  a  horse  or  a  box, 
is  not  an  element  of  number  science — is  not  a  unit;  it 
is  the  idea  one  which  the  mind  forms  upon  viewing  the 
object  as  an  entirety  that  constitutes  the  unit,  or  fun- 
damental element  in  the  science  of  numbers. 

The  student  is  next  asked  to  re-think  the  nine 
general  classes  of  numbers,  to  find  the  basis  of  each 
classification,  and  to  give,  by  definition,  the  mark  of 
each  class. 

The  classification  of  numbers  is  discussed  thus  early 
not  because  of  its  logical  relation  to  that  which 
precedes  or  succeeds  it,  but  because  of  the  basal  char- 
acter which  Number  classification  sustains  in  all  com- 
putation. 

Number  Representation  is  next  discussed.  Under 
the  Arabic  Notation  it  is  observed  that  the  characters 
are  used  to  represent  numerical  values  thought  in 


three  systems  of  numbers.  Each  of  these  systems,  to- 
gether with  its  notation,  is  discussed. 

Number  Reduction  is  next  considered.  The  kinds 
of  reduction  are  determined,  and  applied  to  numbers 
thought  in  the  decimal,  the  fractional  and  the  com- 
pound systems.  Attention  is  called  to  the  fact  that 
reduction  descending  is  effected  by  multiplication,  and 
that  reduction  ascending  is  effected  by  division,  whether 
the  number  to  be  reduced  be  an  abstract  integer,  a 
fraction  or  a  denominate  number. 

Under  Number  Processes,  the  phases  of  synthesis 
and  analysis  are  treated.  The  terms  used,  the  mental 
acts  involved,  and  the  principles  which  guide  the 
mind  in  computing  are  discussed. 

In  formulating  definitions  and  principles  it  is 
sought  in  the  main,  to  "bring  before  the  mind  the  act 
or  process  by  which  the  concept  to  be  defined  is  sup- 
posed to  be  constructed."  It  has  been  a  special  aim  to 
make  every  definition  sufficiently  inclusive  to  embrace 
all  that  the  term  covers  wherever  found  in  the  work: 
e.  g.  The  definitions  of  multiplication  and  division  as 
given  in  most  text  books  on  Arithmetic,  are  but  par- 
tial since  they  do  not  include  multiplication  and  divi- 
sion by  a  fraction.  The  definitions  in  this  book  are  be- 
lieved to  be  ample. 

Both  common  and  decimal  fractions  are  treated 
together,  the  principles  of  the  one  being  the  principles 
of  the  other. 

The  suggestions  given  for  the  treatment  of  Com- 


pound  Numbers,  will,  it  is  hoped,  enable  the  teacher 
to  proceed  more  systematically  and  satisfactorily  than 
by  the  methods  usually  presented. 

Most  of  the  definitions  and  discussions  under  the 
applications  of  Percentage  are  omitted,  not  because 
they  are  unimportant,  but  because  they  are  so  well 
given  in  text  books  on  Arithmetic  that,  their  repeti- 
tion here  is  deemed  unnecessary. 

Methods  of  solving  representative  problems  have 
been  freely  inserted  under  these  applications. 

Involution  and  Evolution  are  treated  Arithmetic- 
ally instead  of  Geometrically  and  in  a  manner  at  once 
simple  and  exhaustive. 

"Results  in  teaching  depend  upon  the  clearness 
with  which  distinctions  are  made;"  and  the  pupil  can 
be  brought  to  make  clear  distinctions,  only  by  being 
held  to  rigidly  logical  modes  of  thinking.  As  an  aid 
to  this  end,  forms  of  solution  are  given  for  nearly  all 
classes  of  arithmetical  exercises.  These  forms  or  others 
equally  logical  should  be  strictly  adhered  to  in  order  to 
secure  to  the  pupil  the  maximum  culture  which  the  subject 
can  give. 

A  number  of  errors,  typographical  and  otherwise, 
were  observed  after  the  work  was  in  print.  Some  of 
these  have  been  corrected  with  the  pen.  Others  still 
remain,  but  they  are  of  such  a  character  as  not  to  mis- 
lead the  attentive  reader. 

TEREE  HAUTE,  APRIL,  1884. 


CONTENTS. 


SECTION  I. 
Number  Genesis.— Page  9 

SECTION  II. 
Number  Classification — Page  12. 


Integer 12 

Fraction 12 

Abstract 12 

Concrete 12 


Prime 12 

Simple .  13 

Denominate 13 

Compound 13 


Composite 12 

SECTION  III. 
Number  Representation.— Page  14. 


Notation 14 

Koman 14 

Arabic 16 

Systems  of  Numbers  ...  16 


The  Decimal  System  . 
The  Fractional  System 
The  Compound  System 


SECTION  IV. 

Number  Reduction. — Page  24. 

Reduction  Descending  .    .    24  |    Reduction  Ascending 


16 
21 
23 


25 


SECTION  V. 

Number  Processes.— Page  26. 

Rules  for  Squaring  ...  35 

Subtraction 38 

Division 40 

Disposition 44 

Divisibility  of  Numbers  .  45 

Evolution  .    . 48 


Computation 26 

Addition 27 


Multiplication 29 

Composition 32 

Involution 34 

Tables  of  Squares  ....  35 


SECTION  VI. 

Measures  and  Multiples.— Page  5O. 

Greatest  Common  Divisor  50  |   Least  Common  Multiple  .    53 

SECTION  VII. 
Fractions.— Page  55. 


The  Primary  Idea  ....  55 
Classes  of  Fractions  ...  56 
General  Principles  .  .  .58 
Reduction  .  .  •  .  ...  59 


Addition  and  Subtraction.  65 

Multiplication 67 

Division 71 

Aliquots 80 


SECTION  VIII. 

Compound  Numbers.— Page  82. 

Diagram  82  I   Application— Time  Mea- 

Order  of  Study 88  |          sure 84 

SECTION  IX. 
Time  and  Longitude.— Page  95. 

SECTION  X. 
Areas  and  Volumes.— Page  1OO. 

SECTION   XI. 
The  Decimal  System  of  Measures.— Page  1O4. 

SECTION  XII. 
Percentage.— Page  112. 


The  Terms  Used  ....  112 

Relations 113 

General  Cases 114 

Applications 11 3 

Profit  and  Loss 118 

Commission 123 

Stocks 127 


Insurance 130 

Taxes , 133 

Customs 134 

Interest 135 

Discount 142 

Exchange 147 


Equation  of  Payments  .  .    148 
SECTION  XIII. 
Ratio  and  Proportion.— Page  ISO. 

Ratio. 150  I   Proportional  Parts.  158 

Proportion  ...    ....    152  j 

SECTION   XIV. 
Involution  and  Evolution.— Page  159. 


Involution  of  Numbers  to 
Second  Power  ...  160 

Evolution  of  the  second 
Root 163 


Involution  of  Numbers  to 
Third  Power  ....    166 

Evolution  of  the  Third 

Root 169 


SECTION  XV. 
Test  Problems.— Page  17O. 

SECTION    XVI. 
Review  Questions  and  Topics.— Page  184. 


OUTLINES 


OF- 


NUMBER  SCIENCE. 


SECTION  I. 
NUMBER  GENESIS. 

1.  Mathematics  may  be  defined  in  a  general  way  as 
the  department  of  knowledge  which  exhibits  the  prop- 
erties and  relations  of  extension  and  number. 

2.  A  consideration  of  extension  leads  into  the  realm 
of  space.     Geometry  and  kindred  branches  are  evolved 
from  the  properties  and  relations  of  extension. 

3.  A  consideration  of  number  leads  into  the  realm 
of  time  for  the  primal  elements  of  the  science. 

The  Basis  of  Number. 

4.  In  General,     a.  Every  conscious  state  of  the  mind 
is  known  to  begin,  to  endure  for  a  period  and  to  end, 
giving  way  to  another  mental  phenomenon. 

6.  Mental  states  are  distinct  from  the  mind  and 
from  one  another.  They  are  known  to  be  distinct  be- 
cause experienced  in  different  periods  of  time  or  be- 
cause of  a  diversity  of  the  states  compared. 


10 

c.  To  give  genesis  to  the  idea  of  number  it  is  nec- 
essary for  a  state  to  succeed  a  state  in  consciousness. 
The  idea  of  number  thus  originates  in  the  cognition 
of    succession.     Succession  is  possible  only  in  time; 
hence  time  is  conditional  for  the  numerical  idea. 

d.  Arithmetic  is  a  branch   of  the  mathematics  of 
number;  it  is  both  a  science  and  an  art. 

e.  As  a  science,  Arithmetic  exhibits  the  facts  of 
number,  presents  the   relations  sustained   among  the 
facts,  formulates  these  relations  into  principles  which 
bring  the  facts  together  until  they  appear  in  conscious- 
ness systematically  arranged  as  an  organic  whole. 

As  an  art,  Arithmetic  is  the  application  of  the 
science  in  computation. 

5.  In  Particular.  If  the  attention  be  directed  to  an 
object,  as  a  tree,  a  house,  an  apple,  etc.,  among  the  at- 
tributes observed  is  that  of  oneness.  The  idea  one  may 
be  abstracted  from  the  conception  of  the  object  which 
occasions  it,  and  as  thus  abstracted,  it  may  be  thought 
apart  from  its  object. 

The  Unit.  The  term  integral  unit,  or  unit  one  (from 
Davies)  is  fixed  upon  to  designate  the  idea  one  as  ab- 
stracted from  the  conception  of  an  object  thought  as  a 
whole,  as  not  composed  of  identical  parts,  nor  itself 
one  of  the  identical  parts  which  compose  another 
whole. 

[In  mathematics  equality  is  identity.  The  reader 
is  referred  to  Everett's  Science  of  Thought,  pp.  98-105, 
inclusive.] 

Remark.  The  primary,  or  integral  one,  arises  in  the  mind  as 
above  stated;  there  are,  however,  two  classes  of  secondary  ones 
with  which  we  shall  have  much  to  do  in  our  investigations  of 
number  relations  and  processes. 


11 

1.  The  Fractional  Unit.    If  an  object  be  separated, 
into  equal  parts,  and  the  attention  be  directed   to  a 
part  thus  found,  the  idea  one  arises. 

Definition.  The  idea  one,  which  is  applicable  to  a 
part  that  results  from  the  separation  of  an  object  into 
equal  parts  is  called  a  fractional  unit. 

2.  The  Multiple  Unit.     If  the  mind  think   sever- 
al objects  as  forming  a  group,  the  idea  one  arises. 

Definition.  The  idea  one  which  is  applicable  to  a 
group  of  wholes  may  be  called  a  Multiple  Unit. 

Remarks.  1.  The  integral  unit,  or  unit  one,  is  the  primary 
idea  in  Arithmetic.  All  other  units  are  definitely  related  to  this 
primary  unit  through  the  objects  from  which  the  units  are 
originally  abstracted. 

2.  An  object  giving  rise  to  the  idea  one  may  be  called  a 
unit-object.    Most  writers  on  Arithmetic  call  such  an  object  a  unit. 

3.  The  object  which  gives  rise  to  a  fractional  unit  is  one  of 
the  equal  parts  into  which  the  unit-object  of  the  integral  unit 
is  thought  as  separated.    In  view  of  this  relation  of  part  to 
whole,  a  fractional  unit  is  often  said  to  be  derived  from  the  in- 
tegral unit  by  the  analysis  of  the  latter  into  equal  parts. 

The  unit,  which  is  the  idea  one,  is  a  unity,  and  is  incapable 
of  division  ;  its  object,  only,  can  be  divided,  a  new  idea  one 
arising  when  a  part  instead  of  a  whole  engages  the  attention. 
[See  Unity  in  Fleming's  Vocabulary  of  Philosophy.] 

A  Number.  A  number  may  be  defined  as  a  unit  or 
group  of  like  units  thought  together. 

Remarks.  1.  The  units  composing  a  number  may  be  integral, 
fractional  or  multiple. 

2.  Aristotle  did  not  include  unity  in  the  idea  of  number. 
He  considered  unity  as  the  element  of  number. 

Locke  included  unity  in  the  idea  of  number.  This  view  is 
adopted  by  modern  writers  on  Arithmetic. 


12 

SECTION  II. 
NUMBER  CLASSIFICATION. 

a.  On  the  basis  of  integral  or  fractional  units  used 
in  their  formation,  numbers  are  classified  as  integers 
and  fractions. 

6.  An  Integer.     A  number  composed  of  one  or  more 
integral  units  is  called  an  integer. 

7.  A  Fraction.     A  number  composed  of  one  or  more 
fractional  units  is  called  a  fraction. 

Remark.  An  integer  and  a  fraction  thought  as  combined  are 
together  called  a  mixed  number. 

6.  On  the  basis  of  application  to  objects  numbers 
are  classified  as  Abstract  and  Concrete. 

8.  Abstract.     A  number  thought  as  independent  of, 
or  separate  from  an  object,  is  called  an  abstract  num- 
ber. 

9.  Concrete.     A  number  thought  as  applied  to  an 
object  is  called  a  concrete  number.     Examples  :  4  men, 
12  horses,  f  of  a  dollar,  etc. 

Remark.  4,  12  and  f,  in  the  examples  are  concrete  num- 
bers because  their  objects  are  named. 

c.  On  the  basis  of  divisibility,  abstract  integers  are 
classified  as  Composite  and  Prime. 

10.  Composite.     A  number  that  can  be  divided  into 
equal  integers  each   greater  than  one,  is  a  composite 
number. 

11.  Prime.     A  number  that  cannot  be  divided  into 
equal  integers  each  greater  than  one,  is  a  prime  num- 
ber. 


13 

d.  On  the  basis  of  distinct  or  assumed  unit-object, 
concrete  numbers  are  classified  a  s  Simple  and  Denom- 
inate. 

12.  '  Simple.     A  number  whose  unit-object  is  a  dis- 
tinct whole,  is  a  simple  number.     As  5  books,  3  pens,  7 
horses;  the  unit-objects  being  the  distinct  wholes,  book, 
pen  and  horse,  respectively. 

Remark.  An  abstract  number  is  often  called  a  simple  num- 
ber. 

13.  Denominate.     A  number  whose  unit-object  is  an 
assumed  time,  extent  or  degree  of  intensity,  is  a  de- 
nominate number.     As  3  days,  5  feet,  7  pounds ;   the 
unit-objects  being  the  day,   the  foot    and  the  pound, 
respectively,  each  of  which  is  an  assumed  unit-object. 

14.  Compound.     Two  or  more  denominate  numbers 
having  the  same  primary  unit,  are  together  called  a 
compound  number. 

Remarks.  1.  A  denominate  number  is  often  defined  as  a 
number  whose  unit  (object)  is  named,  but  such  definition  is 
applicable  to  all  concrete  numbers  and  is,  therefore,  too  inclu- 
sive. 

2.  A  compound  number  is  often  defined  as  a  number  con- 
sisting of  two  or  more  denominations.     Under  this  definition 
5  ft.  3  Ib.  3  hr.  is  a  compound  number.    The  error  in  the  defi- 
nition is  apparent. 

3.  The  classification  of  fractions  is  deferred  until  the  sub- 
ject is  treated  in  detail. 


14 

SECTION  III. 
NUMBER  REPRESENTATION. 

Notation. 

15.  Definition.  Notation  is  a  systematic  method  of 
representing  numbers  by  symbols. 

Kinds  of  Notation. 

Remark  Many  kinds  of  notation  have  been  in  use  in  differ- 
ent times  and  places.  But  two  of  these,  however,  are  usually 
presented  in  Arithmetic,  viz.,  that  used  by  the  ancient  Ro- 
mans and  that  introduced  into  Europe  by  the  Arabs. 


The  Roman  Notation. 

16.  Characters.     The  alphabet  of  the  Koman  nota- 
tion consists  of  the  seven  letters,  viz. :     I.  V.  X.  L.  C. 
D.  M. 

17.  Signification.     I  represents  one,  V  five,  X  ten,  L 
fifty,  C  one  hundred,  D  five  hundred,  M  one  thousand. 

18.  Relation.     \T  represents  five  times  I. 

X  «  two  «  V. 

L  "  five  "  X. 

C  "  two  "  L. 

D  "  five  "  C. 

M  «  two  "  D. 

19.  Limit.     The  Eoman  notation  is  limited  to  the 
representation  of  integers,  and  is  chiefly  used  in  num- 
bering chapters,  headings  and  divisions  in  books  and 
papers. 


15 


20.    Principles. 


Remark.     As  now  used  the  Roman  notation  is  effected  in  ac- 
cordance with  the  following  principles  : 

I.  If  a  letter  be  written    with   its   equal  or  with  a 
combination  of  its  equals,  the  letters  combined  repre- 
sent a  value  equal  to  the  sum  represented  by  the  let- 
ters. 

II.  If  a  letter  be  placed  at  the  left  of  a  letter  rep- 
resenting a  greater  value,  the  letters   combined  repre- 
sent the  diiference  between  the  values  represented  by 
the  letters  taken  separately. 

III.  If  a  letter  or  combination  of  letters  be  placed 
at  the  right  of  a  letter  representing  a  greater  value, 
the  letters   combined    represent    the    sum  of  the  two 
values, 

IV.  If  a  d.ash  be  placed  over  a  letter  or  combination 
of  letters  the  value  represented  is  multiplied  by  one 
thousand. 

21.  Exercises.  Read  each  of  the  following :  IX 
XIV,  XXVII,  LIII,  XCV,  CXXVIII,  XII,  XCVI, 
MDCCL,  MCI,  MDCCCL,  LXXXI1I,  DIV. 

Write  in  the  Roman  notation  :  Twenty-nine ;  thirty- 
three  ;  fourteen  ;  one  hundred  six  ;  fifty-six  ;  one  thou- 
sand two  hundred  sixty -four;  seventy-eight  ;  one  thou- 
sand seven  hundred  seventy-six ;  ninety-seven  ;  forty- 
nine;  five  hundred  thirteen;  eleven  hundred  seventy. 
[Give  additional  oxercises.J 


16 
The  Arabic  Notation. 

22.  Characters.     The  alphabet  of  the  Arabic  nota- 
tion   consists  of  the   following  ten   characters   called 
figures,  viz. :     1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

23.  Signification.      The  first  nine  of  these  figures 
represent  one,   two.   three,    &c.,   units  to  nine.     They 
are  called  significant  figures,    or   digits   because  they 
signify  or  point   out   numbers.     The   tenth   figure   is 
called   zero   or   nought.      It   expresses   no   numerical 
value. 

Remark.  In  Algebra  the  zero  is  classified  among  the  symbols 
of  quantity  and  is  defined  as  the  representative  of  an  infinitely 
small  quantity. 

24.  Systems.     The  Arabic  figures  are  used  in  three 
distinct  systems  of  numbers,   viz.:     (1.)     The   decimal. 
(2.)  ThG  fractional     (3.)  The  compound. 


1.— The  Decimal  System. 

25.  A  Scale.     The  series  of  units  constituting  the 
basis  of  a  system  of  numbers  is  called  a  scale. 

26.  The  Decimal   Scale.      The  decimal  system  of 
numbers  has  for  its   basis   a   series   of  units  each   of 
which  is  ten  times  as  great  as   the   unit   next  below  it 
in  the  series.     This  series  of  units  is  called  the  decimal 
scale. 

27.  Orders  of  Units.     1.  The  integral  unit,  or  unit 
one,  is  called  a  unit  of  the  first  order. 

2.  Ten  units  of  the   first,    or   units'  order   are   to- 
gether called  a  unit  of  the  second  order. 


17 

3.  Ten  units  of  the  second,  or  tens'   order  are  to. 
gether  called  a  unit  of  the  third  order. 

4.  Ten  units  of  the  third,  or   hundreds'    order  are 
together  called  a  unit  of  the  fourth  order. 

5.  Ten  units  of  the  fourth,  or  thousands'  order  are 
together  called  a  unit  of  the  fifth  order. 

6.  Ten  units  of  the  fifth,   or  ten-thousands'   order 
are  together  called  a  unit  of  the  sixth  order. 

Remark.  Higher  orders  of  units  are  formed  by  thinking 
together  ten  units,  respectively,  of  the  sixth,  seventh,  eighth, 
ninth,  tenth,  &c..  orders  indefinitely. 

28.  Periods.  1.  The  first  three  orders  of  units  in 
the  decimal  scale,  viz.:  units',  tens'  and  hundreds',  con- 
stitute a  period  called  units'  period. 

2.  The  three   orders   of  units   next   higher  than 
units'  period,   constitute  thousands'   period.     Units  of 
thousands,  tens  of  thousands  and  hundreds  of  thousands 
are  thought  as  composing  this  period. 

3.  The   three  orders   of  units   next   higher  than 
thousands'  period   constitute   millions'   period.     Units 
of  millions,  tens   of  millions   and  hundreds   of  millions 
are  thought  as  composing  this  period. 

4.  Other  periods  of  three  orders  each  are  formed 
f-1  higher  orders  of  units  in  the  decimal   scale.     The 
periods  above  those  named  are  those  of  billions,  tril- 
lions, quadrillions,  quintillions,  sextillions,   septillions, 
octillions,  nonillions,  decillions,  undecillions,  etc.,  to  in- 
finity. 

5.  Each  period  embraces   three   orders,  viz.:  units, 
tens  and  hundreds  of  that  period. 


18 


29.    Lower  Orders. 


Remarks,  a.  The  decimal  scale  may  include  units  lower  than 
the  unit  one. 

6.  The  division  of  a  unit  into  tenths,  hundredths,  thou- 
sandths, ten-thousandths,  etc.,  is  called  a  decimal  division  of 
the  unit. 

1.  Units  which  result   from   dividing  the  unit  one 
into  ten  equal  parts  are  called  tenths,  or  units  of  tenths' 
order. 

2.  Units    which   result  from    dividing  one-tenth 
into  ten  equal  parts  are   called  hundredths,   or  units  of 
hundredths'  order. 

3.  Units  which  result  from  dividing  one-hundredth 
into  ten  equal  parts  are  called  thousandths,    or  units  of 
thousandths'  order. 

4.  Orders  of  decimal  units  called   respectively  ten- 
thousandths,  hundred-thousandths,  millionths,  ten-millionths, 
etc.,  are  formed  by  dividing  into  ten  equal  parts  the  unit 
next  higher  in  the  scale. 

30^     Principle  of  the  Decimal  Scale.    Ten  units  of 
any  order  make  a  unit  of  the  next  higher  order. 


Notation  of  Numbers  Thought  in  the  Decimal  Scale. 

31.  Characters.      The    Arabic   characters    already 
given,  together  with  a  point  or  period,  called  the  deci- 
mal point,  are  used  in   representing   numbers  thought 
in  the  decimal  scale. 

32.  Signification.     The  digits  and  the  zero  have  the 
signification  already  stated,  while  the  decimal  point  is 
used  to  mark   the  place   in   a   written  number  from 
which  to  start  in  determining  its  value. 


19 

33.  Units'  Place.     The  first  place  at  the  left  of  the 
decimal  point  is  units'  place. 

Remark.  Any  number  of  first  order  units  from  1  to  9,  may 
be  represented  by  writing  the  proper  figure  in  units'  place. 

34.  Tens'  Place.     The  place  next  at  the  left  of 
units'  place,  is  tens'  place. 

Remark.  Any  number  of  second  order  units  from  1  to  9,  may 
be  represented  by  writing  the  proper  figure  in  tens'  place. 

35.  Hundreds'  Place.     The  place  next  at  the  left  of 
tens'  place  is  hundreds'  place. 

Remark.  Any  number  of  third  order  units  from  1  to  9,  may 
be  represented  by  writing  the  proper  figure  in  hundreds' 
place. 

36.  Units   of  thousands,   tens     of  thousands,   and 
hundreds   of  thousands,   respectively,   may  be   repre- 
sented in  the  next  three  places  at  the  left  of  those  al- 
ready named.     In  the  next  three  places  may  be  repre- 
sented, respectively,  units  of  millions,   tens  of  millions 
and  hundreds  of  millions,  etc. 

37.  Tenths,  hundredths  and  thousandths,  etc.,  may 
be  represented  by  figures  written  at  the  right  of  units' 
place  in  places  corresponding  to  tens',  hundreds',  thou- 
sands', etc.,  at  the  left. 

38.  Scale.     The  term  scale  is  used  to  name  the 
series  of  successive  units  which  form  the  basis  of  a  sys- 
tem of  numbers.     In  notation,    however,   the  series  of 
successive  places  in  which  the  different  orders  of  units 
may  be  represented  is  called  a  scale.     This  scale  may 
be  called  the  representative  scale  to  distinguish  it  from 
the  thought  scale  defined  above. 

Remark.  The  simple  value  of  a  figure  is  determined  by  its 
form  alone,  while  the  local  value  of  a  figure  is  determined  by 
both  the  form  of  the  figure  and  the  place  it  occupies  in  the 
representative  scale.  . 


20 

39.  Limit.  The  decimal  system  of  writing  num- 
bers is  limited  to  the  representation  of  integers  and 
such  fractions  as  result  from  a  decimal  division  of  the 
the  unit. 


Numeration  and  Beading  Numbers. 

40.  Numeration.     Naming  the  successive  places  in 
a  written  number  is  called  numeration. 

Remark.  In  numerating  a  number  it  is  both  convenient  and 
customary  to  begin  with  units'  order. 

41.  Reading.     Naming  the   numerical  value  repre- 
sented by  a  written  number  is  called  reading  the  num- 
ber. 

Remarks.  1.  In  reading  a  number  it  is  both  convenient  and 
customary  to  begin  at  the  highest  order  in  which  numerical 
value  is  represented. 

2.  In  reading  a  number  the  word  and  should  never  be  used 
except  between  the  integral  and  the  fractional  parts  of  a 
mixed  number,  Thus  325  is  read,  three  hundred  twenty-five ; 
4f  is  read,  four  and  two  thirds ;  420.005  is  read,  four  hundred 
twenty  and  five  thousandths. 

[Exercise  in  writing  and  reading  numbers  in  the 
decimal  scale.] 

Eead  46;  326;  460;  2346;  1785;  57689;  32567; 
4567890;  45678834;  456784352;  46789716;  456.3;  34; 
51.6;  317.04;  45607;  5678.17;  45.041;  32.117;  2.3456 ; 
4.055,  56.78946  ;  .346  ;  .3 ;  .5678  ;  .56789047  ;  .03456789  ; 
.45678;  .5678789; 

Write  each  of  the  following  in  the  decimal  scale  : — 
Two  hundred  thirty-four ;  seven  thousand  sixty-five; 
Five  hundred  forty-one  ;  seven  thousand  seventy-six  ; 
two  thousand  ninety-eight ;  Four  hundred  thousand 


21 

sixteen;  forty-five   thousand   ten;    sixty-six  thousand 
ninety-four ;  seventeen  thousand  five ;  nineteen  thou- 
sand nineteen  ;   five  hundred  thousand  six ;   six  million 
four  hundred  thousand  seventy-eight;   three  hundred 
million  seven  thousand  six  hundred  nine ;  three  tenths  ; 
nine  tenths;   fifteen    hundredths ;    seven   hundredths; 
fourteen  thousandths  ;  six  thousandths ;   two  ten-thou- 
sandths; three  hundred  eleven  thousandths;  one  hun- 
dred two  ten -thousandths  ;  twenty-four  hundred  thou- 
sandths ;  five  hundred  and  seven  hundred  thousandths; 
sixteen   millionths  ;  four  hundred  sixty-one  millionths  ; 
fifty-seven  ten-millionths ;  fifteen    tenths;  twenty-one 
tenths ;   two  hundred   thirteen   tenths ;    five   hundred 
tenths ;    seventy     tenths ;    one    thousand  seventy-five 
tenths  ;  twenty  tenths  ;  one  hundred  fifteen  hundredths ; 
two  hundred  six  hundredths ;  four  hundred  forty-one 
hundredths;  fifty  hundredths  ;  six  hundred  hundredths  ; 
four  hundred  fifty  hundredths ;  four  hundred  seventeen 
tenths;  throe  thousand  hundredths  ;  four  thousand  two 
hundred  eighty -four  thousandths;  two  thousand  seven 
thousandths  ;  one  hundred  tenths;  forty-five  thousand 
thousandths;   forty   tenthb,     ten   thousand   ten-thou- 
sandths; two  hundred  and  six  tenths;  forty  and  seven- 
teen hundredths  ;  eleven  and  five  thousandths  :  seventy- 
five  and  six  tenths ;   one   thousand   and  sixty-five  ten- 
thousandths. 

2.— The  Fractional   System. 

42.     The  primary  idea  of  a  fraction  is  composed  of 
the  following  elements  : 

a.  The  idea  of  a  whole  divided. 

b.  The  idea  o±  equality  of  the  parts. 

c.  A  number  of  the  parts. 


22 

1.  Two  numbers   are   thus   necessary   to  the  con- 
ception of  a  fraction,     a.  The  number  of  equal  parts 
into  which  the  unit  is  thought  as  separated,     b.  The 
number   of  those   parts   thac   are  thought   as  consti- 
tuting the  fraction. 

2.  These  two  numbers  are  called  the  terms  of  the 
fraction.     The  number  of  equal  parts  into  which  the 
unit,  or  whole,   is  thought  as  separated  is   called  the 
denominator  of  the  fraction.     The  number  of  equal 
parts  thought  as  constituting  the  fraction  is  called  the 
numerator  of  the  fraction. 

3.  Denomination.     The  fractional  denomination  of 
a  fraction  is  the  same  as  the  ordinal  of  the  denomina- 
tor.    This  is  true  of  all  fractions  except  those  having 
the  number  two  for  a  denominator.     The  denomina- 
tion of  such  fractions  is  half  instead  of  second,  the  or- 
dinal of  the  denominator. 


The  Notation  of  a  Fraction. 

43.  Fractions  which  result  from  a  decimal  division 
of  the  unit  may  be  written  in  the  decimal  (represent- 
ative) scale.  [Art.  37.]  Other  fractions  have  a  notation 
peculiarly  their  own. 

44.  Since  two  numbers  are  necessary  to  the  thought 
fraction,  two  written  numbers  are  necessary  to  notate 
a  fraction.     These   written   numbers   have   the  same 
names,  respectively,  as   the  terms  of  the  thought  frac- 
tion which  they  represent. 

45.  a.  The  written  denominator  of  a  fraction  is  the 
figure  or  figures  representing  the  number  of  fractional 
units  into  which  a  unit  is.  thought  as  separated. 


23 

b.  The  written  numerator  of  a  fraction  is  the  fig- 
ure or  figures  representing  the  number  of  fractional 
units  composing  the  fraction. 

46.  The  written  denominator  is  placed  below  the 
written  numerator  and  separated  from  it  by  a  short 
line. 

Remark.  A  fraction  is  thought  as  sustaining  a  definite  rela- 
tion to  the  integral  unit ;  we,  therefore,  think  of  a  written  frac- 
tion as  attached  to  units'  place  in  the  (representative)  decimal 
scale.  If  the  thought  fraction  require  it,  the  written  fraction 
may  be  attached  to  any  other  place  in  the  representative  scale. 

[Exercise  in  writing  and  reading  fractions.] 

3.— The  Compound.  System. 

47.  A  compound  number  is  thought  in  two  or  more 
different  orders  of  units  that  have  the  same  primary, 
or  standard  unit. 

48.  In  compound   numbers  the  number  of  orders 
(denominations)  in  any  "measure"  is  limited  to  the 
number  of  different  unit-objects  agreed  upon  for  meas- 
uring the  attribute  under  consideration. 

49.  The  compound  system  of  numbers  is  based,  not 
so  much  on  the  fact  that  the  several  scales  are  varying, 
as  that  each  "measure"  has  its  own  scale.     Some  of 
these  scales  are  varying,  and  some  of  them  are  uni- 
form.    Each  of  the  common   measures  has  a  varying 
scale,   while   the  "metric"   measures,   including    the 
measure  of  U.  $.  money,  has  a  uniform  and  decimal 
scale.     In  the  old  books  will  be  found  a  duo-decimal 
"measure"  which  is,  of  course,  uniform. 

50.  Each  denominate  number  which  forms  part  of 
a  compound  number,  is   thought  in   the  decimal  sys- 
tem, in  the  fractional  system,  or  in  both. 


24 

The  Notation  of  a  Compound  Number. 

51.  The   denominate    numbers   which    compose   a 
compound  number,  are  written  in  a  descending  series, 
from  left  to  right.      Thus — 4  bu.  3  pk.  5  qt.  1  pt.;  5  da. 
16  hr.47  min.;  I  Ib.  3J  oz.  I  pwt. 

52.  The  names  of  the  orders  or  denominations  may 
be  abbreviated,  but  the  parts  of  a  written  compound 
number  are  not  to  be  separated  by  anjr  mark  of  punc- 
tuation. 

In  reading  a  compound  number  the  word  and 
should  not  be  used  between  any  two  adjacent  denom- 
inate numbers  in  the  series  composing  the  compound 
number. 

Remark.  The  first  compound  number  given  under  Art.  51 
should  be  read — 4  bushels  3  pecks  5  quarts  1  pint.  The  third 
should  be  read — •§•  of  a  pound  3J  ounces  |  of  a  pennyweight. 

[Exercise  in  writing  and  reading  compound  num- 
bers.] 


SECTION  IV. 
NUMBER  REDUCTION. 

53.  Reduction  consists   in  the   change   by  which  a 
given  numerical  value  is  thought  in  another  order  or 
denomination. 

54.  Reduction  Descending.     Reduction   descending 
consists  in  reducing  a  numerical  value  of  any  order  or 
denomination  to  a  lower  order  or  denomination.     It  is 
effected   by  thinking  the  value  of  each    unit  of  the 
given  order  or  denomination  in  the  number  of  units  of 
the  lower   that   are   together   equal  to   a   unit  of  the 
higher. 


25 

55.  Reduction  Ascending.  Reduction  ascending  con- 
sists in  reducing  a  numerical  value  of  any  order  or  de- 
nomination to  a  higher  order  or  denomination.  It  is 
effected  by  thinking  as  a  unit  of  the  higher  the  num- 
ber of  units  of  the  lower  order  or  denomination  that 
are  together  equal  to  a  unit  of  the  higher. 

Exercises. 

Reduce  to  lower  orders  each  of  the  following: 

4  thousands;  14  hundreds;  7  tens;.  3  units;  5  hun- 
dredths ;  4  tenths ;  2  thousandths,  &c. 

Reduce  1,  2,  3,  4,  5,  each  to  3ds,  4ths,  5ths,  6ths,  etc. 

Reduce  £  to  4ths;  f  to  6ths;  to  9ths;  to  12ths;  to  15ths. 

Reduce  J,  f  and  f  each  to  8ths ;  to  12ths;  to  IHths. 

Reduce  f  to  halves ;  f-  to  3ds;  ^  to  halves. 

Reduce  $%  to  4ths ;  %%  to  halves  ;  to  lOths;  to  5ths. 

Reduce  f  to  units ;  f  to  units;  -1/  to  units. 

Reduce  5  days  to  hr.;  to  min.;  to  sec. 

Reduce  1  gal.  to  qt.;  to  pt.;  to  gills. 

Reduce  3  bu.  to  pk. ;  to  qt. ;  to  pt. 

Reduce  128  pt.  to  qt. ;  to  pk. ;  to  bu. 

Reduce  96  gills  to  pt. ;  to  qt.;  to  gal. 

Reduce  500  units  to  tens;  to  hundreds. 

Reduce  .2500  to  thousandths;  to  hundredths;  to 
tenths. 

Reduce  .325  to  hundredths ;  to  tenths. 

Reduce  220  units  to  tens  ;  to  hundreds ;  to  tenths  ; 
to  hundredths. 


26 

SECTION  V. 
NUMBER  PROCESSES. 

Computation. 

56.  Definition.     Computation  consists  in  obtaining 
a  number  or  numbers  from  other  numbers  in  the  light 
of  definite  relations  existing  between  or  among  them. 

57.  The  Mental  Acts  Involved.     (1.)    The  first  act  of 
the  mind  concerned  in  computation  is  an  act  of  com- 
parison.    The  numbers  involved  are  compared  in  re- 
spect of  one  or  more  of  the  following  points,  viz.: 

a.  Concrete  denomination. 

b.  Abstract  denomination  or  order. 

c.  Equality. 

d.  Measurement. 

(2.)  The  mental  act  which  effects  a  computation 
is  an  act  of  synthesis  or  analysis  as  the  conditions  of 
a  given  case  may  require. 

Eemarks.  1.  The  mind  thus  performs  but  three  operations 
upon  numbers,  viz.: 

a.  COMPARISON. 

b.  SYNTHESIS. 

c.  ANALYSIS. 

The  first  of  these  is  preliminary  in  its  nature  while  the  other 
two  are  the  processes  by  means  of  which  all  numerical  compu- 
tation is  effected. 

2.  Synthesis,  as  here  used,  means  putting  together,  while  analysis 
means  separating,  or  taking  apart. 

58.  The  primary  judgment   in  computation   is  a 
proposition  of  identity,^,  e.,  something  equals  something. 
This  judgment  is  called  an  equation. 

The  values  between  which  the  relation  of  equality 
exists  are  called  the  members  of  the  equation. 


27 
The  Synthesis,  or  Combination  of  Numbers. 

i — ADDITION. 

59.  Sum.     The  sum  of  two  or  more  numbers  is  a 
number  equal  in  value  to  them. 

60.  Addition.     Finding  the  sum  of  numbers  is  called 

addition. 

61.  Addends.     The  numbers  to  be  added  are  called 
addends. 

62.  The  Mental  Acts  Involved.     (1.)   The  mind  com- 
pares two  addends  in  respect  of  concrete  denomination 
and  order. 

(2.)  The  mind  begins  with  one  of  the  addends  and 
from  its  number  of  units  counts  until  the  units  of  the 
other  addend  are  used.  The  number  reached  is  the 
sum  of  the  two  addends. 

Remarks.  1.  An  addition  table  is  readily  formed,  the  mas- 
tery of  which  enables  the  mind  to  give  from  memory  the  sum 
of  any  two  addends  within  the  limits  of  the  table. 

2.  With  the  sum  formed  by  the  synthesis  of  two  numbers, 
as  above  indicated,  the  mind  may  combine  another  number, 
and  so  may  continue  the  act  of  combining  one  number  with 
another  so  long  as  there  are  numbers  to  be  added  in  a  given 
case. 

63.  Principles.     I.     Only  like  numbers  can  be  added 
together. 

Remarks.  1.  Like  numbers  are  abstract  numbers  having 
like  units  or  concrete  numbers  having  like  unit-objects.  And 
since  a  collection  of  objects  can  be  named  from  a  common  at- 
tribute only,  so  only  like  numbers  can  be  named  together  or 
added. 

2.  If  unlike  numbers  are  to  be  added  together  a  common 
name  or  denomination  must  be  found  for  them.  They  are  then 
thought  together  under  that  name. 


28 

II.  The  sum  is  of  the  same  order  or  denomination 
as  the  addends. 

III.  The  sum  equals  its  addends. 

IV.  The  sum  exceeds  any  of  its  addends. 

V.  If  the  same  numerical  value   be  added  to  each 
of  two  equal  values,  the  resulting  sums  are  equal. 

64.  The  Sign.  A  perpendicular  cross  (-f-)  placed 
between  two  numbers  indicates  that  they  are  to  be 
added  together.  The  sign  is  read  plus. 


General  Remarks. 

1.  In  adding  numbers  of  any  order  in  the  decimal  scale,  a 
sum  exceeding  9  is  often  found.    Since  such  a  sum  cannot  be 
represented  in  the  place  in  the  written  scale  which  corresponds 
to  the  order  of  units  composing  the  sum,   a  part  or  all  of  the 
sum  must  be  reduced  to  units  of  one  or  more  higher  orders  in 
the  scale  before  it  can  be  uotated.     Reduction  ascending  is  in- 
volved. 

2.  The  definitions,  processes  and  principles  stated  above  ap- 
ply equally  well  whether  the  addends  be  thought  in  the  deci- 
mal, the  fractional  or  the  compound  system  of  numbeis. 


Exercises  in  Addition. 

Remarks.    For  the  exercises  numbered  in  the  left  margin 
read  across  the  page  to  the  right. 

(1)     (2)     (3)     (4)     (5)     (6)  (7)  (8) 

(9)  34     67      47      27      17      16  65  21  * 

(10)  76    98      58      39      19     27  47  18* 

(11)  26    46      74      66      28      94  86  31f 

(12)  38    78      23      72      30      89  75  481- 

(13)  46  96   17   87   48   67  49  59J 

(14)  71  84   29   93   85   86  87 


29 
(15)  (16)  (17)  (18)  (19)  (20)  (21)  (22) 


(23) 

3.5 

26} 

4.6 

5 

.1 

3J 

21* 

4. 

7 

213 

(24) 

4.6 

47i 

3. 

7 

6 

.7 

5* 

46| 

8. 

5 

46.7 

(25) 

7.4 

38£ 

4. 

1 

8 

.5 

16 

23i 

9. 

7 

54.5 

(26) 

3.9 

59| 

5. 

8 

9 

.4 

14 

48^- 

6. 

5 

68.7 

(27) 

6.2 

78|- 

9 

.9 

6 

.3 

15J 

56J 

5.4 

59.9 

(28) 

9.3 

67i 

9.8 

5 

.9 

18|   87*   3.2 

97.5 

t 


(29)  (30)  (31)  (32)  (33)  (34)   (35) 

(36)  412  567  2678  4678  567.83  56789  54347 

(37)  '316  898  4513  8764  387.65  98765  45678 

(38)  578  596  4678  9876  123.45  13759  94352 

(39)  963  347  3542  6789  543.21  24680  24536 

(40)  447  598  2345  8765  234.56  46'802  78579 

(41)  567  678  5436  5678  654.32  20468  45678 

(42)  324  345  7654  7654  345.67  57898  57890 

(43)  568  234  4567  4567  765.43  85765  47875 

(44)  718  432  8765  1234  456.78  45678  32567 

(45)  678  561  5678  4321  876.54  57934  78579 


II — MULTIPLICATION. 

65.  Product.     The   product  of   two  numbers   is  a 
number  that  sustains  the  same  relation  to  one  of  them 
that  the  other  does  to  1. 

Remark.    The  term  multiple  is  sometimes  used  for  product. 

66.  Multiplication.      Finding   the   product   of   two 
numbers  is  called  multiplication. 

67.  Its  Genesis.     (1.)  In  addition  it  is  not  essential 
that  the  addends  be  compared  in  any  respect  except  in 
that   of  denomination    and   order.     They  may,    how- 


30 

ever,  be  compared  in  respect  of  equality.  If  the  ad- 
dends are  equal  their  addition  may  be  called  constant 
addition. 

(2.)  If  the  sum  of  a  constant  addition  be  remem- 
bered so  that  it  may  be  given  when  the  number  of 
equal  addends  and  the  value  of  each  are  known,  the 
act  is  called  multiplication. 

68.  The  Mental  Act.     The  act  of  synthesis  involved 
is  that  of  a  constant    addition   previously  performed, 
while  the  act  called  multiplication  is  but  the  recalling 
of  the  sum  from  memory. 

69.  Multiplicand.     The  multiplicand  is  usually  de- 
fined as  the  number  to  be  multiplied. 

70.  Multiplier.     The  multiplier  is  usually  defined  as 
the  number  by  which  the  multiplicand  is  multiplied. 

Remarks.  1.  If  the  multiplier  be  an  integer  the  multiplicand 
is  one  of  the  addends  of  a  constant  addition  while  the  multi- 
plier is  the  number  of  those  addends. 

f  2.  If  the  multiplier  be  a  fraction  the  process  of  multiplica- 
tion consists  in  obtaining  such  part  of  the  multiplicand  as  the 
multiplier  is  of  1. 

71.  Factors.     The  multiplicand   and  multiplier  are 
called  factors  of  the  product. 

A  factor  of  a  number  is  one  of  its  makers  by  mul- 
tiplication. 

72.  Principles. 

I.  The  multiplicand  may  be   abstract  or  concrete. 

II.  The  product  is  of  the  same  name  and  order  as 
the  multiplicand. 

III.  The  multiplier  is  an  abstract  number. 

IV.  If  both  factors  are  abstract,  they  may  be  used 
interchangeably   without   affecting   the   value   of  the 
product. 


31 

V.  If  either  or  both  factors   be   used   in  parts  the 
sum   of  the   partial   products   thus   found  equals   the 
product  of  the  factors  as  wholes. 

VI.  The  product  sustains  the  same  relation  to  the 
multiplicand  that  the  multiplier  does  to  1. 

VII.  If  the  multiplier  is  1  the  product  equals  the 
multiplicand. 

VIII.  Multiplying  either  factor  by  a  number  mul- 
tiplies the  product  by  that  number. 

IX.  If  two  equal  values  be  respectively  multiplied 
by  the  same  number  the  resulting   products  are  equal. 

X.  A  number  is  multiplied   by  multiplying  one  of 
its  factors. 

Remark.    The  following  principles  are  seen  to  be  true  in  the 
light  of  definitions  yet  to  be  given. 

a.  A  number  expressed   in   the  decimal  system  of 
notation  is  multiplied  by  any  power  of  10  by  removing 
the  decimal  point  as  many  places  to  the  right  as  there 
are  units  represented  by  the  index  of   the  given  power 
of  10. 

b.  Dividing  any  factor  of  a  product   by  a  number 
divides  the  product  itself  by  that  number. 

c.  Dividing  both  factors  of  a  product  by  the  same 
number  divides  the   product   by  the   square   of  that 
number. 

d.  If  one   of  two   factors   be   multiplied   and  the 
other  be  divided  by  the  same  number,  their  product  is 
not  changed. 

73.  The  Sign  of  multiplication  is  an  oblique  cross 
(X).  It  is  read  times  or  multiplied  by,  and  is  placed  be- 
tween two  factors  whose  product  is  required. 


32 

Remarks.  1.  If  both  factors  are  abstract  it  is  immaterial 
which  reading  be  given  to  the  sign ;  if,  however,  one  of  the  fac- 
tors be  concrete,  it  is  thereby  made  the  multiplicand,  [Prin  III,] 
and  if  it  precede  the  sign,  the  sign  is  read  multiplied  by,  while  if 
the  multiplier  precede  the  sign,the  sign  is  read  times. 

2.  If  a  numerical  value  be  expressed  within  a  parenthesis  or 
other  sign  of  aggregation  and  another  number  be  written  with- 
out the  sign  but  not  separated  from  it  by  any  sign,  the  sign  of 
multiplication  is  understood  between  them.    3(4_|_2)=:3X 
(4+2)=l& 

3.  In  effecting  a  multiplication  a  product  exceeding  9  is  often 
found.  Since  such  a  product  cannot  be  repiesented  in  the  place 
in  the  written  scale  which  corresponds  to  the  order  of  units 
composing  the  product,  it  becomes  necessary  to  reduce  to  units 
of  a  higher  order  or  orders  so  much  of  the  product  as  is  thus  re- 
ducible without  involving  fractions.      The  reduction  employed 
is  reduction  ascending. 

[Exercises  in  multiplying  numbers  thought  in  the 
different  systems.] 

in— COMPOSITION. 

74.  The  continued  product  of   several   factors   is 
found  by  multiplying  the  product  of  two  of  them  by  a 
third,  the  product  of  the  three  by  a  fourth,  and  so  on 
until  all  the  factors  are  used. 

75.  A  Composite  Number.     The  product  of  two  in- 
tegral factors,  each   greater  than  1,  or   the  continued 
product  of  several  such  factors  is  a  composite  number. 

76.  Composition.     The  process  of  forming  a  compo- 
site number  is  called  composition. 

77.  A  Prime  Number.     A  number  that    cannot  be 
formed  by  composition  is  called  a  prime  number. 

78.  Numbers  are  relatively  prime  if  they  have  no 
common  factor. 

79.  A  Common  Multiple  of   numbers    is  a  product 
of  which  each  of  them  is  a  factor. 


80.  The  Least  Common  Multiple  of  numbers  is  the 
least  product  of  which  each  of  them  is  a  factor. 

81.  A  Common  Measure  of  numbers  is   a  factor  that 
can  be  used  in  the    formation     of  each  of  them. 

82.  The  Greatest  Common  Measure  of  numbers  is  tLe 
greatest  factor  that  can  be  used  in  the     formation    of 
each  of  them. 

83.  Principles. 

I.  Both  prime  and  composite  factors  may  be  used 
in  composition. 

II.  A  multiple   is   the  product   of  all   its   prime 
factors. 

III.  A  common  multiple  of  numbers   has  in  its 
composition  the  product  of   all  the  prime  factors  of 
each  of  the  given  numbers. 

IV.  The  least  common  multiple  of  numbers  is  the 
product  of  all  the  prime  factors  of  each  of  the  numbers, 
each  factor  being  used  in  the  composition   the  great- 
est number  of  times  it  occurs  in  the  composition  of 
any  one  of  the  numbers. 

V.  A  factor  of  a  number  is  a  factor  of  any  multi- 
ple of  that  number. 

VI.  A   common  factor  of  numbers  is  a  factor  of 
their  sum. 

VII.  The  greatest  common  factor  of  numbers  is 
the  product  of  all  their  common  prime  factors. 

[Exercise  in   forming  composite  numbers,  common 
multiples,  etc.] 


34 


IV — INVOLUTION. 

84.  Power.     A  power  of  a  number  is  the  number 
itself  or  the   product  of  the   number  by  itself  one  or 
more  times. 

85.  Involution.    The  process  of  forming  a  power  is 
called  Involution. 

86.  Root.    One  of  the  equal  factors  used  to  form  a 
power  is  called  a  root. 

87.  Second  Power.     The  product  of  two  equal  fac- 
tors is  called  the  second  power,  or  square  of  either  of 
them. 

88.  Second  Root.     Either  of  the  two  equal  factors 
that  compose  a  second  power  is  called  the  second,  or 
square  root  of  the  given  power. 

89.  Higher  powers  are  formed  by  the  composition 
of  a  greater  number  of  equal  factors.     Every   such 
power  is  named  by  the  ordinal  of  the  number  of  equal 
factors  used  in  the  composition  of  the  power. 

90.  A  root  is  called  the  third,  fourth,  etc.,  if  it  be 
one  of  three,  four,  etc.,  equal  factors  that  compose  a 
power. 

91.  Any  number  is  called  both  the  first  power  and 
the  first  root  of  itself.     A  first  power  can  enter  into  a 
synthesis  with  its  equal  and  thus  become  a  component 
of  a  higher  power  but,  is  itself  not  formed  by  involu- 
tion. 

92'  The  index  of  a  power  is  a  small  symbol  of 
number  written  at  the  right  and  above  a  given  number 
and  indicates  that  the  number  is  one  of  as  many  equal 
factors  as  there  are  integral  units  expressed  by  the 
index.  Thus  23=8,  62^36,  etc. 


35 
93.     Table  of  Squares  from  I2  to  252. 


2«=    4 

32^    9  152=225 

41=  16  162-=256 

&=  25  172=289 

62_  36  182=324 

72=  49  192=361 

32^  64  202=400 

92^  81  212=--441 

10^=100  222 


25*=626 
94.    Table  of  Cubes  from  I3  to  93. 


23=  8 

33=27  ?3; 

43=64  83=512 

93=729. 

95.     Rules  for  Squaring  Numbers. 

I.  To  Square  a  Number  ending  in  £. 

1.  Square  the  integer. 

2.  Add  \  the  integer. 

3.  Add  -fa 

II.  To  Square  a  Number  ending  in  \. 

C  i.  Square  the  integer. 
a  I  2.  Add  the  integer. 

(  3.  Add  i. 

6.  Multiply  the  integer  by  the  integer  next  greater 
and  add  J. 


III.  To  square  a  number  ending  in  £ . 

1.  Square  the  integer. 

2.  Add  f  of  the  integer. 

3.  Add  ^. 

IV.  To  square  a  number  ending  in  5. 

1.  Square  the  tens. 

2.  Add  the  tens. 

3.  Annex  25. 

6.  Multiply  the  simple  value  of  the  tens  by  the 
number  next  greater  and  annex  25. 

V.  To  square  a  number  between  25  and  50. 

1.  Take  25  from  the  number. 

2.  Take  the  difference  from  25. 

3.  Square  the  remainder. 

4.  Add  the  first  difference  as  hundreds. 

VI.  To  square  a  number  between  50  and  75. 

1.  Take  50  from  the  number. 

2.  Add  the  difference  to  25. 

3.  Call  the  result  hundreds. 

4.  Add  the  square  of  the  difference. 

VII.  To  square  a  number  between  75  and  100. 

1.  Take  the  number  from  100. 

2.  Take  the  difference  from  the  number. 

3.  Call  the  result  hundreds. 

4.  Add  the  square  of  the  first  difference. 

VIII.  To  square  a  number  ending  in  25. 

1.  Square  the  hundreds. 

2.  Add  J  the  hundreds. 

3.  Call  the  result  ten-thousands. 

4.  Add  625. 


37 
i 

IX.  To  square  a  number  ending  in  75. 

1.  Square  the  hundreds. 

2.  A.dd  f  of  the  hundreds. 

3.  Call  the  results  ten-thousands. 

4.  Add  5625. 

X.  To  multiply  a  number  of  two  orders  by  11. 

Think  the  sum  of  the  terms  of  the  multiplicand 
between  them  : 

As— 34  X  11=374.     54x  11=594. 
[Exercises.] 

96.     Remarks  on  Synthesis. 

1.  Numerical  synthesis  classifies  itself  under  four  heads,  viz  , 
addition,  multiplication,  composition  and  involution. 

There  is,  however,  but  a  single  method  of  synthesis,  and  that 
is  addition.  In  multiplication,  including  composition  and  in- 
volution, a  mm  is  remembered,  this  sum  having  been  previously 
found  by  the  synthesis  called  addition. 

2.  In  addition  two  numbers  are  combined  by  one  impulse  of 
the  mind  without  regard  to  the  equality  of  the  numbers. 

3.  In  multiplication  a  given  number  of  equal  numbers  may 
be  thought  as  combined  at  once.    The  number   of  equal  num- 
bers is  not  limited  to  two,  but  may  be  any  number  whose  sum, 
found  by  constant  addition,  can  be  given  immediately  from 
memory. 

This  remark  holds  only  with  an  integral  multiplier. 

4.  In  composition  a  definite  number  of  factors  are  used  in 
continued  multiplication. 

5.  In  involution  a  definite  number  of  equal  factors  are  used 
in  continued  multiplication. 


38 

The  Analysis,  or  Separation  of  Numbers. 

Remark.  Since  a  number  may  be  obtained  by  adding  to- 
gether any  two  parts  which  form  it,  it  follows  that  the  number 
may  again  be  resolved  into  its  parts. 

I — SUBTRACTION. 

97.  Difference.     The  difference   between  two  num- 
bers is  a   number   which    added   to   one   of  them  will 
make  a  sum  equal  to  the   other. 

Remark.  In  Arithmetic  the  term  difference  may  be  denned 
as — the  numerical  excess  of  one  number  over  another. 

98.  Subtraction.     Finding  the    difference   between 
two  numbers  is  called  subtraction. 

Remark.  Subtraction  may  be  denned  as — taking  a  part  of  a 
number  from  the  number. 

99.  The  Minuend.     The  sww  involved  in  subtraction 
is  called  the  minuend. 

Remark.  The  minuend  is  often  denned  as — the  number  to 
be  diminished  by  the  withdrawal  of  a  part  of  it. 

100.  The   Subtrahend.      The  known  part  of   the 
minuend  is  called  the  subtrahend. 

Remarks.  1.  The  subtrahend  is  usually  defined  as  the  num- 
ber to  be  subtracted. 

2.  The  term  remainder  is  often  used  to  designate  the  part  of 
the  minuend  that  is  left  after  the  withdrawal  of  the  subtra- 
hend. The  number  so  designated  is,  however,  the  difference 
between  the  minuend  and  subtrahend. 

101.  The  Mental  Acts.     1.  If  the  sum  of  two  num- 
bers and  one  of  them  be  known,  the  other  is  found  by 
the  process  called  subtraction.     The  mental  act  con- 
sists in  presenting  the  required  part  from  memory,  or 
it  may  consist  in  counting  from  the  given  part  to  the 
given  sum. 


39 

2.  If  the  difference  between  two  separate  numbers 
be  required,  the  mind  compares  the  two  numbers  in 
respect  of  concrete  denomination  and  order.  If  the 
numbers  are  found  to  be  similar,  the  mind  proceeds  to 
withdraw  or  think  away  from  the  greater,  a  part  which 
equals  the  less.  The  number  remaining  is  the  excess 
of  the  greater  over  the  less,  and  is,  therefore,  their 
difference. 

102.  The  Sign.     The  sign  of  subtraction  is  a  single 
dash,  ( — )  placed  after  the  minuend  and  before  the 
subtrahend  in  an  indicated  subtraction. 

103.  Principles. 

I.  Only  like  numbers  are  used  in  subtraction. 

II.  The  sum   of   the   subtrahend  and  difference 
equals  the  minuend. 

III.  If  the  minuend  and  subtrahend  be  equally 
increased  the  difference  between  the   yums  thus  ob- 
tained equals  the  difference  between  the  minuend  and 
subtrahend. 

IV.  If  the   same  value  be  taken  from  two  equal 
values  the  remainders  are  equal. 

V.  It  either  or  both  minuend  and  subtrahend  be 
used  in  parts,  the  partial  remainders  combined  equal 
the  entire  remainder. 

Remarks.  1.  Since  the  minuend  is  a  sum,  it  is  greater  (in 
Arithmetic),  than  the  subtrahend.  It  sometimes  occurs,  how- 
ever, that  there  is  a  less  number  ol  units  of  some  order  in  the 
minuend  than  of  the  same  order  in  the  subtrahend.  In  such 
case  the  minuend  must  be  prepared  before  the  subtraction  can 
be  effected.  This  preparation  consists  in  reducing  a  unit  of  the 
order  next  higher  in  the  minuend  to  units  of  the  required  or- 
der and  combining  them  with  the  units  of  that  order.  If  there 
be  no  units  of  the  order  next  higher  in  the  minuend,  the  work 
of  reduction  must  begin  at  the  first  order  up  the  scale  in  which 


40 

numerical  value  is  thought.  A  unit  of  that  order  is  reduced 
to  units  of  the  next  lower ;  one  of  the  units  resulting  from  this 
reduction  is  then  reduced  to  units  of  the  order  next  lower,  and 
so  on  until  the  number  of  units  of  each  order  in  the  minuend 
equals  or  exceeds  that  of  each  order  in  the  subtrahend.  The 
subtraction  is  then  readily  effected.  The  reduction  involved  is 
reduction  descending. 

2.  The  reduction  mentioned  in  remark  1,  may  be  avoided  by 
effecting  the  subtraction  in  the  light  of  Principle  III,  adding 
10  units  of  the  deficient  order  in  the  minuend  to  the  units  of 
that  order,  and  then  compensating  this  addition  by  adding  1 
unit  of  the  next  higher  order  to  the  subtrahend. 

[Exercises  in  Subtraction.] 

II — DIVISION. 

104.  Quotient.     The  quotient  of  one  number  by  an- 
"other  is  a   number  that   sustains  the  same  relation  to 

the  first  number  that  1  does  to  the  second. 

Remark.  The  quotient  of  one  number  by  another  is  the  fac- 
tor which,  used  with  the  second  number,  will  produce  the  first. 

105.  Division.  Finding  a  quotient  is  called  division. 

Remark.  In  the  light  of  remark  under  Quotient,  division 
may  be  defined  as  finding  one  of  two  factors  of  a  given  product 
when  the  other  factor  is  known. 

106.  Dividend.     The  number  to  be  divided  is  called 
the  dividend. 

Remark.  The  dividend  is  the  given  product  of  which  the 
divisor  is  the  known  factor. 

107.  Divisor,     The  term  divisor  is  usually  defined 
as  the  number  by  which  the  dividend  is  divided. 

Remark.  I.  The  divisor  is  the  factor  given  with  the  dividend 
to  determine  the  quotient. 

2.  A  particular  problem  may  be—Given  the  dividend  and 
quotient  to  find  the  divisor.  This  probiem  is  solved  by  divid- 
ing the  dividend  by  the  quotient.  The  quotient  of  a  preceding 
division  thus  becomes  the  divisor  in  the  given  problem. 

108.  Remainder.     The  term  remainder  is  applied  to 
a  part  of  a  given  dividend  that  may  remain  undivided 
in  any  case. 


41 
The  Genesis  .of  Division. 

109.  (1.)  If  a  given  subtrahend  be  taken  from  a 
given  minuend,  and  again  be  taken  from  the  remain- 
der, and  again  be  taken  from  the  second  remainder, 
and  so  on  until  the  given  minuend  is  exhausted  or  gives 
a  remainder  less  than  the  constant  subtrahend,  the 
several  subtractions  viewed  together  are  called  a  con- 
stant subtraction. 

(2.)  If,  when  the  minuend  and  constant  subtra- 
hend are  known  the  mind  gives  from  memory  the 
number  of  subtractions  necessary  to  exhaust  the  given 
minuend;  or,  if  the  minuend  and  the  number  of  sub- 
tractions that  can  be  made  are  known,  and  the  mind 
gives  from  memory  the  constant  subtrahend,  the  act  is 
called 


(3.)  If,  when  a  product  arid  one  of  two  factors  that 
produce  it  are  known,  the  mind  gives  from  memory 
the  other  factor  the  act  is  called  division. 

110.     The  Mental  Act. 

(1.)  The  mental  act  which  effects  a  division  is  the 
presentation  (from  memory)  of  the  quotient  when  the 
dividend  and  divisor  are  known. 

(2.)  The  act  may  be  a  memorized  constant  sub- 
traction; or,  it  may  be  the  recalling  from  memory  of 
one  of  two  factors  when  the  other  and  their  product 
are  known. 

Remark.  In  effecting  the  division  of  a  decimal  or  a  com- 
pound number,  if  the  divisor  be  numerically  greater  than  the 
number  of  unitp  of  the  order  or  denomination  to  be  divided,  the 
latter  number  must  be  reduced  to  units  of  a  lower  order  or  de- 
nomination before  the  division  can  be  effected  without  involv- 
ing a  fractional  quotient.  Reduction  descending  is  involved. 


42 

III.     Principles. 

I.  The  product  of  the  divisor  and  quotient  equals 
the  dividend. 

II.  If  the  dividend  be  divided  in  parts  the  sum  of 
the  several   partial  quotients   obtained   is  the  entire 
quotient. 

III.  If  the  dividend  be  divided  by  the  factors  of 
the  divisor  used  in  continued  division,  the  final  quo- 
tient is   the  quotient  of  the  dividend  by  the  entire 
divisor. 

Remark.  In  applying  this  principle  a  remainder  may  occur 
upon  dividing  by  one  or  more  of  the  factors  of  the  divisor. 
These  partial  remainders  do  not  constitute  the  ultimate  or  true 
remainder. 

The  following  is  a  method  for  determining  the  true 
remainder. 

Example. — Divide  1377  by  294,  using  the  prime  fac- 
tors of  the  divisor,  and  determine  the  true  reniainder. 
Solution. — The  prime  factors  of  294  are  2,  3,  7,  7. 


2 

Partial 
Remainders 

|  1377        

True 
Remainders. 

8 

I    688  

1 

7 

I    229                           1 

2 

~j      32  5 

30 

~~4...                       4  . 

168 

201 

Explanation. 

a.  1.  The  entire  dividend  divided  by  2  gives  a 
quotient  of  688  and  a  remainder  of  1. 

2.  688,  which  is  approximately  \  the  entire  divi- 
dend, divided  by  3  gives  a  quotient  of  229  and  a  re- 
mainder of  1. 


43 

3.  229,  which    is   approximately   J  of  the   entire 
dividend,  divided  by  7  gives  a  quotient  of  32  and  a  re- 
mainder of  5. 

4.  32,  which   is   approximately  -^   of  the  entire 
dividend,  divided  by  7  gives  a  quotient   of  4  and  a  re- 
mainder of  4. 

5.  4,  the   final   quotient   is   approximately 
the  entire  dividend,  or  the  part  required. 

b.  1.  If  1  remain  upon  dividing  £  the  dividend, 
2  times  1,  or  2,  would  remain  upon  dividing  the  entire 
dividend. 

2.  If  5  remain  upon  dividing  £   of  the  dividend,  6 
times  5,  or  30  would  remain   upon   dividing  the  entire 

dividend. 
* 

3.  If 4  remain  upon   dividing^  of  the   dividend, 
42  times  4,  or  168,  would  remain  upon  dividing  the  en- 
tire dividend. 

4.  We  thus   find  that   upon   dividing   the   entire 
dividend   we   should    have   remaining   l+2-|-30-}-168j 
or  201,  as  the  ultimate,  or  true  remainder. 

The  accuracy  of  this  result  may  be  tested  by  di- 
viding the  entire  dividend  by  the  divisor  as  a  whole. 

IV.  The  quotient  sustains  the  same  relation  to  the 
dividend  that  1  does  to  the  divisor. 

V.  If  the  divisor  is  1  the  quotient   equals  the  divi- 
dend. 

VI.  If  a  division  require  the  number  of  times  that 
one  number  is   contained   in  another,   the  divisor  and 
dividend  are  like  numbers  and  the  quotient  is  abstract. 


44 

VII.  If  a  division  require  one  ot  the  equal  parts  of 
a  number,  the  dividend  and  quotient    are  like  numbers 
and  the  divisor  is  abstract. 

VIII.  If  two  equal  numerical  values  be  divided  by 
the  same  number,  the  resulting  quotients  are  equal. 

112.  General  Principles  of  Division. 

Remark.     The  following  six  principles  are  called  gener<d  prin- 
ciples : 

I.  Multiplying  the  dividend    by   any  number  mul- 
tiplies the  quotient  by  that  number. 

II.  Multiplying  the  divisor  by  any  number  divides 
the  quotient  by  that  number. 

III.  Multiplying  dividend  and  divisor  by  the  name 
number  does  not  change  the  quotient. 

IV.  Dividing  the  dividend  by  any  number  divides 
the  quotient  by  that  number. 

V.  Dividing  the  divisor  by  any  number  multiplies 
the  quotient  by  that  number. 

VI.  Dividing   dividend   and    divisor   by  the  same 
number  does  not  change  the  quotient. 

[Exercises.] 

Ill DISPOSITION. 

113.  Definition.     The  analysis  of  a   composite  num- 
ber into  its  factors  is  called  disposition,  or  factoring. 

Remarks.     1.  Disposition  is  a  phase  of  division  viewed  as  the 
process  of  finding  the  factors  that  compose  a  multiple. 

2.  Disposition  is  the  reverse  of  composition.     In  composition 
the  factors  are  given  to  find  their  product,   while  in  disposition 
the  product  is  given  to  find  its  factors. 

3.  In  disposition  the  factors  are  found  by  dividing  the  given 
multiple  by  any  exact  divisor  of  it.     The  quotient  thus  found 


45 

is  divided  by  any  exact  divisor  of  itself.  The  second  quotient 
is  divided  by  any  exact  divisor  of  itself,  etc.,  until  the  required 
factors  are  found.  The  several  divisors  used  and  the  final  quo- 
tient are  the  factors  of  the  given  multiple. 

If  the  prime  factors  are  required,  the  several  divisors  used 
and  the  final  quotient  must  be  prime  numbers.  * 

114.  Principles. 

I.  If  a  number  is  divisible   by   two  or  more  num- 
bers  in   continued   division,    it   is   divisible    by   their 
product. 

Remarks.  1.  A  number  is  said  to  be  divisible  by  another  if 
the  quotient  is  an  integer. 

This  is  a  limited  meaning  of  the  word  divisible.  In  the  light 
of  the  definitions  and  principles  of  division,  any  number  is 
divisible  by  any  other  number. 

2.  In  disposition  a  number  is  not  considered  as  a  factor  of 
itself,  nor  is  1  considered  as  a  factor  of  a  number. 

II.  A  common  divisor  of  two  numbers  is  a  divisor 
of  their  difference. 

III.  If  a  number  be  divided    by  one  of  its  prime 
factors  or  by  the  product  of  two  or  more  of  them,  the 
quotient  is  the  remaining  prime   factor  or  the  product 
of  the  remaining  prime  factors  of  the  number. 

IV.  If  the  product   of  two   factors   be  divided  by 
either  of  them  the  quotient  is  the  other. 

V.  If  the  product  of  more   than    two  factors  be  di- 
vided by  one  of  them,  the   quotient   is   the   product  of 
the  other  tactors  of  the  number. 

115.  Divisibility  of  Numbers. 

Remarks.  1.  There  is  no  general  method  devised  whereby 
the  factors  of  a  multiple  may  be  readily  found  ;  nor  is  there 
any  means  whereby  a  number  is  known  to  be  composite.  Cer- 
tain numbers,  however,  possess  characteristic  marks  denoting 
that  they  are  composite.  A  few  of  these  will  be  discussed. 

2.  A  number  whose  units'  figure  is  0,  2,  4,  '5  or  8,  is  called  an 
even  number.  All  other  numbers  are  called  odd  numbers. 


46 

I.  An  even  number  is  divisible  by  2. 
Elucidation.  The  units'   figure  of  every  integer  is 

one  of  the  ten  Arabic  characters.  If  the  number  rep- 
resented  by  any  of  these  figures  be  multiplied  by  2  the 
units'  figure  of  the  product  is  0,  2,  4,  6  or  8. 

II.  If  the  sum  represented  by   the  digits  of  a  num- 
ber be  divisible  by  3,  the  number  is  a  multiple  of  3. 

Elucidation. 


9—  J    500=5  X  100=5(  99+l)=5x  99+5. 
^—  1      10=1  X     10=1(     9+l)=lx     9+1. 
t       2=  2. 

Upon  separating  any  number,  as  4512,  into  parts 
as  indicated  above,  it  is  observed  that  the  last  member 
of  each  of  the  continued  equations  is  separated  into 
two  addends.  The  first  of  each  of  these  parts  is  seen 
to  be  a  multiple  of  3.  [Prin.  V.  page  33.]  The  other 
parts,  together  with  the  second  member  of  the  last 
equation,  are  represented  by  the  several  digits  of  the 
given  number.  If,  therefore,  the  sum  represented  by 
these  digits  be  divisible  by  3,  the  given  number  is  di- 
visible by  3.  [Prin.  VI.  page  33.] 

III.  A  number  is  divisible  by  4  if  its  two  right 
hand  figures  are  zeros  or  represent  a  multiple  of  4. 
Why? 

IV.  A  number  is  divisible    by  5  if  its  units'  figure  is 
0  or  5.     Why  ? 

V.  A  number  is  divisible    by  6  if  it  be  even  and  a 
multiple  of  3.     Why  ? 

VI.  A  number  is  divisible   by  7  if  once  its  units-\- 
3   times    its  tens  +2   times   its   hundreds-}-^   times  its 
thousands-}-^    times     its    ten  -thousands  +5    times     its 


47 

hundred'thousands+tbe  numbers  represented  by  the 
succeeding  figures  multiplied,  respectively,  by  the  se- 
ries of  multipliers  named  above,  be  a  multiple  of  7. 

VII.  A  number  is  divisible  by  8  if   its  units',  tens' 
and  hundreds'  figures  are  zeros  or  represent  a  multiple 
of  8.     Why  ? 

VIII.  A  number   is   divisible  by   9  if  the   sum  of 
the  numbers  represented   by   its   digits   be  a  multiple 
of  9. 

Remark.     This  may  be  elucidated   in   a  manner  similar  to 
that  given  for  divisibility  by  3. 

IX.  A  number  is  divisible  by  10  if  its  units'  figure 
is  0.     Why  ? 

X.  A  number  is  divisible  by  11,    if  the   difference 
between  the  sum  of  the   numbers  represented  by  the 
digits  in  the  odd  places  and  the   sum   of  the  numbers 
represented  by  the  digits  in  the  even  places  is  nothing 
or  a  multiple  of  1.1. 

XI.  A  number  is  disvisible  by  12  if   it  be  a  multi- 
ple of  3  and  4.    Why? 

116.     General  Remarks, 

1.  A  number  is  prime  if  it  fail  of  division  upon  being  tested  by 
every  prime  number  up  to  a  divisor  that  gives  a  quotient  less 
than  the  divisor.     Why  ? 

2.  A  table  of  prime  numbers  in  a  given  series  of  natural 
numbers— as  from  1  to  100,  may  be  formed  by  checking  off  as 
composite  every  second  number  from  2,  every  third  number  from 
3,  every  fifth  number  from  5,   every    seventh   number  from  7, 
every  eleventh  number  from  11,   etc ,  to  the  required  limit. 
The  numbers  remaining  are  prime.    The  series  of  numbers 
with  the  composite  numbers  thus  expunged,  is  called  the  seive 
of  Eratosthenes. 

3.  In  factoring  a  number  the  pupil  should   always  test  it  by 
the  conditions  herein  given,  and  not  guess  at  its  factors  until 
the  tests,  as  far  as  known,  have  been  applied. 

4.  Pupils  should  learn  the  prime  factors  of  every  composite 
number  from  4  to  100. 


48 
IV— EVOLUTION. 

117.     Definition.     The   analysis   of  a  power  into  the 
equal  factors  which  compose  it  is  called  evolution. 

(1.)  Each  of  the  equal  factors  found  by  evolution 
is  called  a  root  of  the  power  from  which  it  is  evolved. 

(2.)  A  root  is  called  the  second,  third,  fourth,  etc., 
according  as  it  is  one  of  two,  three,  four,  etc.,  equal 
factors  that  compose  a  power. 

Any  number  is  called  the  first  root  of  itself. 

(3.)  The  index  of  a  root  is  a  fractional  unit  writ- 
ten at  the  right  and  a  little  above  a  written  power  and 
indicates  by  its  denomination  the  root  required. 

(4.)  The  radical  or  root  sign  (;/)  is  often  used  to 
indicate  a  root.  If  used  alone  before  a  number  it  in- 
dicates the  second,  or  square  root.  If  a  root  other  than 
the  second  is  required,  the  radical  sign  has  placed 
above  it  the  denominator  of  the  fractional  unit  which 
denotes  the  required  root.  8/*=f/8=2. 

(5)  Exponent.  The  index  of  a  power  or  of  a  root 
is  often  called  an  exponent. 

(6.)  If  a  number  be  affected  by  a  fractional  expo- 
nent other  than  a  fractional  unit,  the  ordinal  of  the 
numerator  of  the  exponent  is  the  power  to  which  the 
number  is  to  be  involved,  while  the  ordinal  of  the  de- 
nominator is  the  root  to  be  evolved  from  that  power: 
or,  the  ordinal  of  the  denominator  of  the  exponent  is 
the  root  to  be  evolved  from  the  given  number,  while 
the  ordinal  of  the  numerator  is  the  power  to  which 
that  root  is  to  be  involved.  .  Thus,  8^  indicates  that 


49 

the  third  root  of  the  second  power  of  8  is  required  ;  or 
it  indicates  that  the  second  power  of  the  third  root  of 
8  is  required.     In  either  case  the  result  is  4. 
[Exercises.] 

118.     Remarks  on  Analysis. 

1.  The    analysis    of    numbers    classifies  itself   under   four 
phases,  viz.:  SUBTRACTION,  DIVISION,  DISPOSITION  and  EVOLUTION. 

2.  In  subtraction  a  number  is  thought  as  separated  into  two 
parts  without  regard  to  the  relative  value  of  the  parts. 

3.  In  division  a  number  is  thought  as  separated  into  a  defi- 
nite number  of  equal  parts. 

4.  In  disposition  the  factors  of  a  given  composite   number 
are  found  by  continued  division. 

5.  In  evolution  one  of  a  definite  number  of  equal   factors 
which  compose  a  power  is  found. 

6.  Any  root  is  readily  found  by  reversing  the  stops  taken 
in  forming  the  corresponding  power. 


SECTION  VI. 

APPLICATIONS. 

Remarks.     1.     In  the  preceding  discussion  all  the  processes 
and  general  principles  involved  in  Arithmetic  are  presented. 

The  topics  which  remain  are  but  applications  of  what 
has  already  been  given.  In  some  instances  new  terms  will  be 
used,  but  no  new  process  and  no  new  principles,  except  subor- 
dinate principles,  will  be  found. 

2.  Great  care  should  be  given  to  both  the  "written  form" 
and  the  "thought  form"  for  the  solution  of  examples  under  the 
various  topics  which  follow.     In  many  cases  the  thought  form 
is  so  short  and  simple   that  the  written  form  almost  or  fully 
expresses  it.    In  such  cases  but  one  form  is  given — usually  the 
written  form. 

3.  The  teacher  should  insist  that  the  work  be  placed  neatly 
upon  the  blackboard      It  will  be  found  good  discipline  to  re- 
quire all  members  of  a  class  to  apply  the  same  form  in  the  so- 
lution of  similar  problems.    If  pupils  are  weak,  but  a  single 
form  should  be  given  for  the  same  class  of  exercises.    If  strong, 
pupils  may  master  several  methods  for  doing  the  same  work. 


50 
Measures  and  Multiples. 

GREATEST  COMMON  DIVISOR. 

119.  Definition.     The   greatest   common    divisor  of 
given  numbers  is  the  greatest  number  that  is  contained 
an  integral  number  of  times  in  each  of  them. 

120.  Principles. 

I.  The  product  of  all   the    common   prime  factors 
of  given  numbers  is  their  greatest  common  divisor. 

II.  A  divisor  of  a  number  divides  any  multiple  of 
it. 

III.  A  common    divisor  of  two  numbers   divides 
their  difference. 

IV.  A  common   divisor  of  given  numbers  divides 
their  sum. 

121.  Methods  of  finding  g.  c.  d. 

(1.)  In  the  light  of  principle  I,  we  find  the  prime 
factors  of  the  given  numbers  and  take  the  product  of 
those  factors  that  are  common  to  all  the  numbers.  This 
product  is  the  greatest  common  divisor  of  the  num- 
bers. 

(2.)  The  common  factors  of  given  numbers  may  be 
found  by  dividing  them  by  a  number  that  is  seen  to  be 
a  common  factor  of  them.  Divide  the  resulting  quo- 
tients in  the  same  manner,  and  so  continue  until  quo- 
tients are  obtained  that  are  relatively  prime. 

The  divisors  used  are  the  common  factors  sought 
and  their  product  is  the  required  greatest  common 
divisor.  [Prin.  I.] 


51 

Example.     Find  the  g.  c.  d.  of  24,  30  and  42. 
Written  form. 

2)24  ...30 42. 

3)12.... 15....  21" 

4 5   ..     7. 

2x3=6=g.  c.  d. 

(3.)  The  "division"  method  is  effected  by  dividing 
the  greater  of  two  numbers  by  the  less  (comparing  but 
two  of  the  numbers  at  a  time);  and  then  dividing  the 
divisor  by  the  remainder,  and  so  continuing  to  divide 
the  last  divisor  by  the  last  remainder  until  there  is  no 
remainder.  The  last  divisor  is  the  g.  c.  d.  of  the  two 
numbers  compared.  This  divisor  is  then  compared 
with  another  oi  the  given  numbers  (if  there  be  more 
than  two,)  by  division  as  above  indicated,  and  so  on 
until  the  numbers  in  any  given  case  are  all  used.  The 
last  divisor  thus  found  is  the  g.  c.  d.  of  the  several 
given  numbers. 

Example.     Find  the  g.  c.  d.  of  260  and  716. 
Written  form.  Thought  form. 


260 

196 

64 

64 


16 


716 
520 

196 
192 


Since  260  is  the  greatest  divisor  of  itself, 
if  it  divide  716,  then  ib  260  the  g.  c,  d.  of 
itself  and  716.  Upon  trial  we  find  that 
260  does  not  divide  716,  but  that  the  great- 
est multiple  of  260  in  716  is  520.  By  prin. 


II  we  know  that  the  g.  c.  d.  will  divide  520,  and  by 
prin.  Ill,  we  know  that  it  will  divide  196,  the  differ- 
ence between  716  and  520.  196  is  the  greatest  divisor 
of  itself.  Now  if  it  will  divide  260,  it  will  also  divide  2 
times  260-)- 196,  or  716,  and  hence  will  be  the  required 
g.  c.  d. 


52 

Upon  trial  we  find  that  196  does  not  divide  260, 
but  by  prin.  Ill  we  know  that  the  g.  c.  d.  will  divide 
the  difference  between  260  and  196,  which  is  64.  64  is 
the  g.  d.  of  itself.  Now  if  it  will  divide  196  it  will  di- 
vide the  sum  of  itself  and  196,  or  260,  [Prin.  IV.]  and 
also  2  times  260-J-196,  or  716,  and  hence  will  be  the 
required  g.  c.  d.  64  does  not  divide  196,  but  the  great- 
est multiple  of  64  in  196  is  192.  By  prin.  II  we  know 
that  the  g,  c.  d  will  divide  192,  and  by  prin.  Ill  we 
know  that  it  will  divide  the  difference  between  196  and 
192  which  is  4.  4  is  the  greatest  divisor  of  itself.  Now 
if  4  divide  64  it  will  divide  3  times  64,  or  192,  and  also 
the  sum  of  4  and  192  which  is  196,  and  also  196+64,  or 
260;  and  also  2  times  260+196,  or  716,  and  hence  is 
the  required  g.  c.  d.  4  divides  64,  hence  4  is  the  g.  c. 
d.  of  26d  and  716. 

Remark.  The  work  may  often  be  shortened  by  taking  the 
multiple  of  the  divisor  that  is  nearest  the  value"  of  the  divi- 
dend, and  finding  the  difference  between  that  multiple  and  the 
dividend  for  a  new  divisor. 

In  the  above  example  the  multiple  of  260  that  is  nearest  the 
value  of  716  is  780  and  the  difference  between  780  and  716  is 
64.  We  thus  see  that  the  g.  c  d.  of  the  given  numbers  maybe 
found  by  two  divisions  instead  of  three  as  in  the  solution 
given. 

[Exercises  ] 

Find  the  g.  c.  d,  of  the  following  : 

1.  35,  21,  9,  14.  9.  576  and  168. 

2-  24,  16,  12, 20.  10.  121  <  495. 

3.  63, 27,  36.  11.        21  <•  77. 

4.  20,  42,  54.  12.  260  "  416. 

5.  15,  40,  55,  60, 75.      13.  125  "  500 

6.  64,  72, 100,  96.  14.  294  "  472. 

7.  75,  135,  400.  15.  108  "  146. 

8.  72,  85,  132.  16.  1245  "  600. 


53 

Least  Common  Multiple. 

122.  Definition.     The    least    common    multiple    of 
given  numbers  is  the  least   product   of  which  each  of 
them  is  a  factor. 

123.  Principle.         The   least   common    multiple   of 
numbers  is  the  product  of  all  the  prime  factors  of  each 
of  the  numbers,    each   factor   being  used   the  greatest 
number  of  times  that  it  Occurs  in  any  one  of  the  num- 
bers. 

Example.     Find  the  1.  c.  m.  of  12,  15  and  25. 
Written  form.  Thought  form. 


12—2X2X3. 
15=3X5. 


S^ tains  the  prime  factors  of 

'•  c-  m- 12  which  are  2,  2  and  3- 


Since  the   required  1.  c. 
m.   contains  12,   it   con- 


These  we  take  as  factors  of  the  1.  c.  m.  Since  the  1.  c. 
m.  contains  15,  it  contains  the  prime  factors  of  15 
which  are  3  and  5.  We  have  3  as  a  factor  of  the  1.  c.  m. 
so  we  take  5  as  one  ot  its  factors,  and  thus  have  the 
prime  factors  of  15  as  factors  ot  the  1.  c.  m.  Since  the 
1.  c.  m.  contains  25,  it  contains  the  prime  factors  of  25 
which  are  5  and  5.  We  have  one  5  as  a  factor  of  the 
1.  c.  m.,  so  we  take" another  5  as  one  of  its  factors  and 
thus  have  the  factors  of  25  as  factors  of  the  1.  c.  m. 
We  now  have  as  factors  of  the  required  1.  c.  m.  all  the 
prime  factors  of  12,  15  and  25  and  no  other  factor. 
The  product  of  these  factors=300,  the  required  1.  c.  m. 

Remark..  If  numbers  are  not  readily  factored,  their  1.  c.  m. 
may  be  found  by  either  of  the  following  methods  The  first 
comparison  instituted  in  each  method  is  between  two  of  the 
gi"en  numbers.  Next  between  the  1.  c.  m.  of  the  two  numbers 
already  compared  and  the  third  number.  Next  between  the 
1.  c.  m.  of  the  first  three  numbers  and  the  fourth,  and  so  on 
until  all  the  numbers  in  any  given  case  are  used. 


54 

(1.)  If  one  of  the  two  numbers  be  divided  by 
their  g.  c.  d.  the  quotient  contains  those  factors  of  the 
number  divided  that  are  not  found  in  the  other  num- 
ber. Now  if  the  undivided  number  be  multiplied  by 
this  quotient,  the  product  contains  all  the  prime  factors 
ot  the  two  numbers  and  no  other  factor,  and  hence  is 
their  1.  c.  m. 

A  rule  for  this  method  may  be  formulated  thus  : 
Divide  one  of  two  numbers  by  their  g.  c.  d.  and  multiply  the 
other  number  by  the  quotient.  The  product  is  the  I  c.  m.  of 
the  two  given  numbers. 

(2.)  Since  the  1.  c.  m.  of  two  numbers  contains  all 
the  factors  of  one  of  the  numbers  and  such  factors 
of  the  other  as  are  not  found  in  the  first,  if  the  two 
numbers  be  multiplied  together,  the  common  multiple 
obtained  is  greater  than  their  1.  c.  m.  by  a  factor  equal 
to  their  g.  c.  d 

Hence  the  product  of  two  numbers,  divided  by  their 
g.  c.  d.  equals  their  I.  c.  m. 

[Exercises.] 

Find  the  1.  c.  m. 

1.  8,  14,  18.  7.  24,  32.  42,50. 

2.  21.  16,  36.  8.  15,  28,  40,  65. 

3.  18,  36,  44.  9    6,  8,  10,  12,  14. 

4.  12,  28,  54.  10.  9,  14,  15,  16,  20. 

5.  64,  84. 100.  11.  84.  76  90,  120. 

6.  32,  75,  108.  12.  121,  200  324. 


55 

SECTION  VII. 
FRACTIONS. 

124.  A  Fractional  Unit.     The  idea  one  which  is  ap- 
plicable to  one  of  the  equal  parts  into  which  a  whole 
maybe  thought  as  separated, is  called  a  fractional  unit. 

125.  A  Fraction.     (1.)  A  fraction  is  a  fractional  unit 
or  a  number  of  like  fractional  units  thought  together. 

(2.)  A  fraction  is  one  or  more  of  the  equal  parts 
of  a  unit. 

Remarh.  1.  The  primary  idea  of  a  fraction  is  composed  of 
the  following  elements,  viz. : 

a.  The  idea  of  a  whole  divided. 

6.  The  idea  of  equality  of  the  parts. 

c.  A  number  of  the  parts. 

From  this  analysis  of  the  primary  idea  it  is  apparent  that 
the  value  of  a  fraction  cannot  exceed  that  of  the  unit  which  is 
applicable  to  the  whole  whence  the  fraction  is  derived. 

The  second  definition  is  based  upon  the  primary  idea  as  thus 
analyzed. 

2  Each  of  many  like  units  may  be  thought  as  separated  into 
the  same  number  of  equal  parts  and  any  number  of  these  parts 
may  be  viewed  together  ;  hence  the  first  definition. 

3.  The  expression  J^.  is  not  to  be  interpreted  as  representing 
ten  fourths  of  1,  for  me  object  can  be  thought  into  but  four 
fourths.  JJL  is  to  be  viewed  as  expressing  a  synthesis  of  ten 
fractional  units  each  of  which  is  one  fourth  of  1. 

126.  The  Unit  of  a  Fraction. 

Definition  The  unit,'  or  whole  which  is  thought  as 
divided  into  equal  parts  is  called  the  unit  of  the  frac- 
tion. 

127.  The  Notation  of  a  Fraction.     [See  page  22.] 


56 

Classes  of  Fractions. 

128.  Rased  on  the  decimal  or  non-decimal  division 
of  the  unit  one,  fractions  are  classified  as  decimal  and 
common. 

129.  Decimal.     A  fraction  whose  fractional  unit  is 
a  decimal    part  of  the  unit  one,  is   called  a  decimal 
fraction. 

Remarks.  1.  A  decimal  fraction  is  usually  expressed  by  the 
decimal  notation,  though  it  may  be  written  in  the  fractional 
form. 

2.  A  fraction  expressed  partly  in  the  decimal  and  partly  in 
the  fractional  notation  is  called  a  complex  decimal.  Examples, 
•  3>£;  .034i 

130.  Common.     A  fraction  whose  fractional  unit  is 
other  than  a  decimal  part  of  the  unit  one,  is  called  a 
common  fraction. 

131.  This  classification   of  fractions  is  often  based 
upon  the  method  of  notation   used  in  expressing  the 
fractions;    those   expressed  in  the  decimal   notation 
being  called  decimal  fractions,  and  those  expressed  in 
the  fractional  form  being  called  common  fractions. 

132.  With  1  as  a  basis,  or  standard  of  comparison, 
fractions  are  classified  us  proper  and  improper. 

133.  Proper.     A  fraction  whose  value  is  less  than  1 
is  called  a  proper  fraction. 

134.  Improper.     A  fraction  who^e  value  equals  or 
exceeds  1  is  called  an  improper  fraction. 

Remarks.  1.  The  primary  idea  of  a  fraction  is  a  number  of 
the  equal  parts  of  a  unit. 

The  classification  of  fractions  as  proper  and  iir. proper  is  thus 
seen  to  be  inconsistent  with  the  primary  idea  of  a  fraction. 

2  The  definition  of  a  fraction  as  a' number  of  like  fractional 
units  is  based  upon  a  secondary  idea,  viz  :  A  number  of  like 
parts  of  any  number  of  like  units.  Under  this  definition  any 
number  of  like  fractional  units  is  a  fraction. 


57 

3.  The  two  definitions  of  a  fraction  that  we  have  given  [Art. 
125,]  are  fairly  representative  of  the  definitions  found  in  tlie 
text  books  on  Arithmetic.  Under  neither  of  these  defini- 
tions is  Jiere  ground  for  classifying  fractions  as  proper  and  im- 
proper. 

135.  On  the  basis  of  form,  fractions  are  classified  as 
simple,  compound  and  complex. 

136.  Simple.     A  simple  fraction  is  defined  as  a  frac- 
tion each  of  whose  terms  is  a  single  integer. 

137.  Compound.     A  compound  fraction  is  defined  as 
a  fraction  of  a  fraction. 

\ 

Remark.  The  simplification  of  a  so-cailed  compound  frac- 
tion is  effected  in  the  light  of  principles  which  govern  the  mul- 
tiplication of  one  fraction  by  another,  f  of  f  does  not  express 
a  class  of  fractions,  but  simply  indicates  that  |  is  to  be  multi- 
plied by  f.  [See  Kemark  2,  under  Art.  70.] 

138.  A  complex  fraction   is  defined  as  a  fraction 
having  a  fraction  in  one  or  both  of  its  terms. 

Remarks.  1.  A  complex  fraction  is  read  as  an  expression  of 
division  :  e.  g.  |^  is  read  2£  -5-  3f  and  ^  is  read  \  -5-£. 

In  the  light  of  the  definition  of  denominator,  the  first  of  the 
above  so-called  complex  fractions  is  derived  from  the  division 
of  a  unit  into  3|  equal  parts.  The  mind  sees  at  once  the  im- 
possibility of  such  a  division.  An  examination  of  the  second 
so-called  complex  fraction  given,  renders  still  more  manifest 
the  absurdity  of  calling  these  numerical  values  fractions. 

2.  Since  a  simple  fraction  is  expressed  by  a  form  in  distinction 
from  compound  and  complex,  it  disappears  upon  the  disappear- 
ance of  these  as  classes  of  fractions. 

3.  If  it  is  found  convenient  to  use  the  terms  proper,  im- 
proper, simple,  compound  and  complex  to  distinguish  certain 
phases  of  fractional  notation  or  indicated  processes,  it  may  be 
well  to  retain  those  terms;  but  they  are  certainly  not  necessary 
to  name  any  of  the  essential  thought  elements  of  a  fraction. 


58 

139.     General  Principles. 

Remark.  A  fraction  may  be  considered  as  a  case  of  division. 
The  numerator  being  the  dividend,  the  denominator  the  divisor 
and  tho  fraction  itself  the  quotient.  The  general  principles  of 
division  become,  therefore,  the  general  principles  of  fractionsby 
a  change  of  terminology. 

I.  If  the  numerator  be  multiplied  the  fraction  is 
multiplied  by  the  same  number. 

II.  If  the  denominator  be  multiplied  the  fraction 
is  divided  by  the  same  number. 

III.  If  both  terms  of  a  traction  be  multiplied  by 
the  same  number  greater  than  1,  the  fraction  is  reduced 
to  smaller  fractional  units  but  is  not  changed  in  value. 

IV.  If  the  numerator  be  divided  the  fraction  is 
divided  by  the  same  number. 

V.  If  the  denominator   be  divided  the  fraction  is 
multiplied  by  the  same  number. 

VI.  If  both  terms  of  a  fraction  be  divided  by  the 
same  number  greater  than  1,  the  fraction  is  reduced  to 
larger  fractional  units,  but  is  not  changed  in  value. 

VII.  Fractions  having  a  common  denominator  are 
to  each  other  as  their  numerators. 

Remark.  This  principle  may  be  thus  stated  :  Like  parts  of 
numbers  are  to  each  other  as  the  numbers  themselves,  e.  g. 
the  relation  of  f  to  J  is  the  same  as  that  of  2  to  4  ;  i.  e.  $  of  2 
bears  the  same  relation  to  £  of  4  that  the  whole  of  2  bears  to 
the  whole  of  4. 

VIII.  The  numerator  is  as  many  times  the  value 
of  the  fraction  as  there  are  units  in  the  denominator. 

Remark.  This  principle  is  seen  in  the  relation  of  a  fraction 
to  division.  The  numerator  is  dividend  and  hence  is  the  pro 
duct  of  the  denominator  (divisor)  and  the  fraction  (quotient.) 
A^product  is  as  many  times  either  of  the  two  factors  which 
compose  it  as  there  are  units  in  the  other. 


59 

140.  Reduction  of  Fractions. 

Remarks.  1.  For  definitions  and  kinds  of  reduction,  seepage 
24. 

2.  In  the  light  of  general  principles,  I,  II,  IV,  V,  pupils  will 
readily  multiply  or  divide  a  fraction  by  an  integer. 

The  signs  v  and  /.  are  convenient  to  use  in  some  of  the 
written  forms.  The  former  is  read  "since"  or  "because,"  and 
the  latter  is  read  "hence"  or  "therefore." 

141.  Reduction  Descending. 

CASE  I. 
An  integer  or  mixed  number  to  a  fraction. 

Example.  Reduce  3  to  8ths. 
Writtenform 

3=f-  Thought  form. 

3x8    24 

T\/8~~s'       3=f;  and  by  multiplying  both  terms  of 

f  by  8  we  have  ^. 
Hence  3=V 


Remark.  In  reducing  a  mixed  number  to  a  fraction,  the  inte- 
ger is  reduced  to  the  denomination  of  the  fractional  part  by 
the  above  method  and  the  fractional  part  is  then  added. 

142.    Other  Forms. 

Example.    Reduce  3  to  8ths. 

=3'times    =4. 


C    *•'  l=f» 

a]       3=3  t: 
(    /.3=V. 


1=8  times  £, 

3—8      " 


I    1=|. 
I    8=V- 


Remarks  on  c.    1.  Make  1  =  f  >  and  then  multiply  both  mem- 
bers of  the  equation  by  3.    We  thus  find  that  3  =  ¥• 


60 

2.  In  any  similar  example  make  1  the  first  member  of  the 
first  equation  ;  and  for  the  second  member  take  the  equivalent 
of  1  in  fractional  units  of  the  required  denomination.  Next 
multiply  both  members  of  the  equation  by  the  integer  to  be  re- 
ed. 


__ 
HX4     12 


Exercises.  Perform  each  of  the  following  reductions : 
8  to  15ths;  19  to  4ths ;  8  to  7ths;  6  to  3rds;  5  to  halves. 
[Give  additional  exercises.] 

Reduce  each  of  the  following  to  a  fraction ;  10J ; 
14fc  5i;  3£;  4£;  7J;  3§;  2.5;  3.04;  5.2;  6.7. 

143.     CASE  II. 

A  fraction  to  smaller  fractional  units. 
Example.     Reduce  §  to  12th  s. 
Written  form.  Thought  form. . 

In  the  light  of  principle  III,  §  may  be 
reduced  to  12ths  by  multiplying  both  its 
terms  by  such  a  number  as  will  produce  12  for  the  de- 
nominator of  the  resulting  fraction. 

Both  terms  off  multiplied  by  4=^-.  .-.  %=-£%. 
Example  2.  Reduce  1  to  a  decimal  fraction. 

Written  form. 

7  X  125^^75  =  875  Thought  form. 

8X125     1000  ln  the  light  of  Prin.  Ill,  *  is  re- 

duced to  a  decimal  by  multiplying  both  its  terms  by 
such  a  number  as  will  produce  a  decimal  denominator 
for  the  resulting  fraction.  Both  terms  of  f  multiplied 
by  125-,%%.  /.  1=.875. 


Remarks.  1.  A  decimal  fraction  may  be  transferred  from  the 
fractional  to  the  decimal  notation  by  omitting  the  written  de- 
nominator and  supplying  the  decimal  point. 

2.  A  decimal  fraction  may  be  transferred  from  the  decimal 
to  the  fractional  notation  bv  writing  the  proper  denominator 
under  the  written  decimal  and  removing  the  decimal  point 
from  the  written  numerator. 


61 

3.  The  change  of  notation  thus  effected  is  not  properly  a  re- 
duction, for  the  size  and  number  of  fractional  units  in  the  frac- 
tion remain  unchanged. 

4.  If  a  decimal  fraction  is  to  be  reduced  to  smaller  decimal 
units,  the  application  of  the  above  form  is  best  shown  by  ex- 
pressing the  given  decimal  in  the  fractional  notation. 

144.     Other  Forms. 

Example.  —  Reduce  I  to  12ths. 


1=2  times  A=A- 
.'.  *=-*• 

Written  form  Thought  form. 

For  the  first  equation   we  take 
l=|f  .     Equation  (2)  is  found  by 


dividing  both  members  of  equation  (1)  by  3.  Equation 
(3)  is  obtained  by  multiplying  both  members  of  equa- 
tion (2)  by  2.  We  thus  find  that  f  = 


Example  2.     Reduce  .2  to  thousandths. 

p.  -1=1.000, 

.1=.!  of  1.000=.100; 
c' }  and  .2=2  times  .100=.200. 
t    /.  .2=.200. 

[Exercises.] 

Perform  the  following  reductions,  writing  all  the 
decimal  fractions  in  the  decimal  notation: 

I  to  tenths  ;Jg-  to  24ths;  f  to27ths;  f  to  21st  s;. 5 
to  lOOths  ;  .5  to  lOOOths; f  to  15ths  ;  3i  to  4ths;  3.2  to 
lOOths;  f  to20ths;  f  to  lOOths;  f  to  ISths;  3.2  to 
lOOOOths ;  |  to  54ths. 

[Give  additional  exercises.] 


62 

145.     CASE  III. 

Fractions  to  a  common  and  to  the  least  common 
denominator. 

Remark.  This  case  is  one  of  reduction  descending  in  which 
the  reduction  is  effected  as  in  Case  II. 

Definitvms.  a.  Fractions  have  a  commoYi  denom- 
inator if  composed  of  like  fractional  units. 

b.  Fractions  have  their  least  common  denomina. 
tor  if  composed  of  the  greatest  possible  like  fractional 
units. 

Remarks.  1.  Before  reducing  fractions  to  equivalent  frac- 
tions having  the  1.  c.  d.  it  is  necessary  that  the  terms  of  each 
fraction  be  relatively  prime. 

2.  In  case  II,  the  denominator  of  the  fraction  resulting  from 
the  reduction  is  a  multiple  of  the  denominator  of  the  given 
fraction.  From  the  nature  of  the  method  employed  in  effect- 
ing the  reduction,  this  must  always  be  the  case  ;  hence  a  com- 
mon denominator  of  fractions  must  be  a  common  multiple  of 
the  denominators  of  the  given  fractions. 

Principle.  The  least  common  denominator  of  given 
fractions  is  the  1.  c.  m.  of  their  denominators. 

Exaufple.     Reduce  f ,  f  and  -|  to  equivalent  fractions 
having  their  least  common  denominator. 
Thought  form. 

(1.)  Prin.  The  1.  c.  d.  of  given  fractions  is  the 
1.  c.  m.  of  their  denominators. 

(2.)  The  1.  c.  m.  of  3,  4  and  6  is  12,  hence  each  of 
these  fractions  must  be  expressed  in  12ths. 

(3.)  In  the  light  of  general  principle  III,  each  of 
these  fractions  may  be  reduced  to  12ths  by  multiply- 
ing both  its  terms  by  such  a  number  as  will  produce  12 
for  the  denominator  of  the  resulting  fraction. 

(4.)  Both  terms  off  multiplied  by  4=-^. 
«        «  f  «  3—-. 


63 

(5.)  /.  f,  I  and  £  reduced    to    equivalent   fractions 
having  their  1.  c.  d.  equal,  respectively,  T8Y,  T9^  and  i-|. 

Remark.  Examples  in  this  case  may  be  solved  by  stating  (1) 
and  (2)  as  in  the  above  thought  form,  and  then  reducing  to  the 
required  denomination  by  any  of  the  forms  given  under 
Case  II. 

[Exercises.] 

Reduce  to  a  common  and  1.  c.  d.  the  following  : 

if  t;  *,*,*;! f  t;  *,if;  i,  M;  M»  .4,  ^; 

f ,  .5,  i  ;  .06,  |,  i,  .05  ;  f ,  M>  t ;  2 J,  f ,  f,  4i ;  3|,  2.5, 5J. 


146,     Reduction  Ascending. 
CASE  I. 

A   fraction   to   an    integer  or   a   mixed   number: 
or  to  the  decimal  scale. 

Example  1.  l-£-—  what  integer  or  mixed  number  ? 

Written  form.  Thought  form. 

17-^-5 3f 02.  I      Since  a  fraction  is  reduced  to  larger 

5 -=-5  1  F  I  units  by  dividing  both  its  termfc  by 
the  same  number,  we  may  reduce  y  to  integral  ones 
by  dividing  both  its  terms  by  5.  The  result,  3f-f-l,=3£. 

Remark.     Dividing  both  terms  of  a  fraction  by  its  denomina- 
tor reduces  the  fraction  to  the  denomination  of  the  unit  1. 


Example  2.     Eeduce  -£fo  to  a  decimal  fraction. 
Written  form.  Thought  form. 

—~*~^==——=  08 
400--4     100 


is  reduced    to   the  denomina- 
tion hundredths    by  dividing  both 
its  terms  by  4;  the  result,  ^^=.08,  a  decimal  fraction. 

Remark.  The  reduction  of  a  common  fraction  to  a  decimal 
fraction  may  be  effected  by  either  reduction.  The  guiding 
thought  is — to  multiply  or  divide  both  its  terms  by  such  a  num- 
ber as  will  give  a  decimal  denominator  to  the  resulting  fraction. 


64 

147.    Other  Forms. 

Example  1.     y=  what  integer  or  mixed  number? 

f  U=17— 5. 
a.  I  17—5=31. 


!>.  1  y==as  many  1's  as  5  is  contained  times  in  17,  or  3f. 


Example  2.     Reduce  f  to  the  decimal  scale. 
Form, 


Remark.  The  denominator  of  a  decimal  fraction  is  a  power 
of  10,  the  prime  factors  of  which  are  2  and  5.  It  follows,  there- 
fore, that  if  any  fraction  (in  its  lowest  terms)  have  in  its  de- 
nominator a  factor  other  than  those  of  10,  such  fraction  cannot 
be  reduced  to  a  decimal.  A  common  fraction  is  reducible  to  a 
decimal  if  the  denominator  of  the  given  fraction  is  divisible  by 
2  or  5,  or  by  2  and  5  and  by  no  other  prime  number.  If  a  fraction, 
in  its  lowest  terms,  have  in  its  denominator  a  prime  factor 
other  than  those  ot  a  decimal  denominator,  a  repetend  or  cir 
culate  will  result  upon  attempting  to  reduce  the  fraction  to  the 
decimal  scale. 

'  Exercises.     Reduce  to  integers  or   mixed  numbers 

the  following  : 

JL6  •  JLJL  •     21  •      37    •    _2  0  •     3.  •    i  •      3_9  •    4JL  .      2.5  .    JM 5  .  100  . 
5     >     6     >      3'        5     >      ¥    >      2  >     3  >      13?      7     '        11    ^T  '       3     ' 

2.5.3.. 

Reduce  each  of  the  following  to  the  decimal  scale 
or  to  a  complex  decimal. 

11.    13.  J.5.  •     3-7.     9.1.  .2.1.      5..     1.      2-      3.     1     .       5     . 
T  (f  >   16'   16'  TTJ"  >   25  '  ^TT  >3'37°?     6'T>     T'     T  '   1 2"  '    T2"  ' 

1     .     3     .     1    •     2    •     6 
1'S  '  T3  '    15?   15?    16' 


148.     CASE  II. 

A  fraction  to  larger  units,  or  lower  terms. 

Example.     Reduce  -|  to  thirds. 

Written  form.  Thought  form. 

6~3_2  I  In  the  light  of  Prin.  VI,  f  may  be  re- 
9-i-3  3  I  duced  to  3rds  by  dividing  both  its  terms 
by  such  a  number  as  will  give  3  for  the  denominator  of 
the  resulting  fraction.  Both  terms  off  divided  by  3, 


Remark.  A  fraction  is  reduced  to  its  lowest  terms  by  divid- 
ing both  its  terms  by  their  g.  c.  d. 

Exercises.     Reduce  each  of  the   following  to  lower 
and  lowest  terms. 

**'    16.144-  10  f.  •     1006-      325.      117     .       7     •       8.25.48- 

3TJTFT2'  3T ~5  '     4TRT4  '      6TO~'TT46>     "2"!  >     "SIT  >    4~0~  )    ^~0~  ? 
1246 •     144 
T6"8T>    1728' 

149.  General  Remarks  on  Reduction  of  Fractions. 

1.  General  principle  Hi  provides  for  the  reduction  of  a  frac- 
tion to  smaller  units;  while    general  principle  VI  provides  for 
the  reduction  of  a  fraction  to  larger  units. 

2.  From  the  foregoing  presentation  of  reduction  of  fractions 
we  find  that  the  several  cases  are  readily  classified  under  one  or 
the  other  of  the  two  kinds  of  reduction  ;  and  that  each  ca&e  in- 
volving reduction  descending    may  be  effected  in  the  light  of 
general  principle  III,  while  each  case  involving  reduction  as- 
cending may  be  effected  in  the  light  of  general  principle  VI. 

150.  Addition  and  Subtraction  of  Fractions. 

Principle — Only  like  units  can  be  added. 

Principle — Only  like  units  can  be  used  in  subtrac- 
tion. 

Remark.  In  adding  or  subtracting  fractions  it  is  to  be  ob- 
served that  the  numerators  are  the  numbers  with  which  we  deal 
while  the  denominators  only  give  name  to  those  numbers. 


Example.     Add  f,  £  and  f. 

Solution.  Since  only  like  units  can  be  added, 
these  fractions  must  be  reduced  to  a  common  denom- 
inator before  adding. 

Written  form. 

•    2  __  8    3  _  9  5  __  10 

T—  12;   4  ~12"     [  6~W 
_8_+J__|_10_=2^==2i 
12      12  ^12      12 


Remarks.     1.  A  form  for  subtracting  one  fraction  from  an- 
other is  similar  to  that  for  addition. 

2.  If  mixed  numbers  are  to  be  added  or  subtracted,  the  inte- 
gers and  the  fractions  may  be  operated  upon  separately  and  .the 
results  combined,  or,  the   mixed  numbers  may  be  reduced  to 
fractions  and  the  result  found  by  the  form  given  for  examples 
in  addition,  and  by  a  similar  form  for  examples  in  subtraction. 

3.  It  is  not  essential  that  fractions  expressed  in   the  decimal 
notation  be  reduced  to  the  same  fractional  denomination  be- 
fore adding  or  subtracting.     It  is  to  be  observed  that  only  like 
orders  of  units  can  be  added  or  subtracted. 

Exercises. 
(1.)  Add  |,  f,  f  ;  f,  i,  A;  |,  «,  A,  i;  |,  *,  i,  *,  f  ; 

T\,  |;  .2,  4,  i,  |;  3%,  ^  ^;  f  ^,  2,  5,  6*. 


(2.)  Perform  the  operation  indicated  in  each  of  the 
following  : 

f-i;i-f;  f-i;  A  -I;  A-  A;  3.4  -ij; 

^  -  f;  2TV  -  A;  3.15  -  A;  6J  -TV;  25.04  -  21.002; 
364.006  —  162.1;  32.5  —  1.0001. 


67 
Multiplication  of  Fractions. 

151.     CASE  I. 
To  multiply  by  an  integer. 

Example  1.  Multiply  T\  by  3. 
Thought  form.     Since  a  fraction  is   multiplied   by 
multiplying  its  numerator,  f\X3=^f. 

Example  2.  Multiply  fV  by  2. 
Thought  form.     Srnoo  a  fraction  is  multiplied  by 
dividing  its  denominator, 


Remark.  .  If  the  product  should  not  be  in  its  lowest  terms  or 
if  it  be  reducible  to  an  integer  or  mixed  number  it  should  be 
reduced. 

[Exercises.] 

Multiplicands:^]  f;  f  ;  f;  T2^;  ^;  T45;  fV;  84;  jfo 
4J;  51;  15fc  21f. 

Multipliers  :    4,  5,  3,  2,  6,  7,  8,  10,  12,  15. 


152. 

To  multiply  by  a  fraction. 

Example  1.  Multiply  6  by  * 

[6X1=6. 

6=     of  6=2. 
First  farm  -{ 


$=2  times  2=4. 

I  The  multiplier  =j  of  2, 
6X2=12. 
6Xiof  2=4  of  12=4. 
'L/.  6XI-4.' 


Principle.  The  product  sustains  the  same 
relation  to  the  multiplicand  that  the  multi- 
plier does  to  1. 

Third  form        In  this  examPle  tn^  multiplier  is  §  of  1  ; 
hence  the  product  is  f  of  6. 
if  of  6=2. 

\  of  6=2  times  2=4. 
>.  6XI-4. 

r 

[Give  principle  and  statement  as  in  third 
form.] 

Fourth  form  §  of  6=J  of  2  times  6,  or  12, 
4-  of  12=4. 


V 


Written  form. 
(1.)     6=6. 
(2.)     §X3=2. 
(3.)     6X1X3=6X2. 
(4.)     6X1=2X2=4. 
.-.  6XI-4. 

Thought  form. 
6=6. 

Also  the  multiplier  multiplied  by  its  de- 
\  nominator  equals  its  numerator. 

Multiplying  together  the  corresponding 
members  of  these  equations  gives  equation 
(3). 

Dividing  both  members  of  equation  (3)  by 
3,  gives  equation  (4). 

The  first  member  of  equation  (4)  consists 
of  the  two  factors  whose  product  is  required, 
while  the  second  member  is  the  result  sough  t. 


Fifth  form( 


Example  2.  Multiply  f  by  f 

Remark.  The  forms  of  solution  for  this  example  do  not  dif 
fer  from  those  given  for  example  1.  Those  forms  are,  however, 
applied  to  the  solution  of  example  2. 


ix  1=1. 


f     The  multiplier  is  |  of  5. 


T/iird  form 


Principle.  The  product  sustains  the  same 
relation  to  the  multiplicand  that  the  mul- 
plier  does  to  1. 

In  this  example  the  multiplier  is  f  of  1 ; 
hence  the  product  is  f  of  f. 

I  of  !=A- 

f  of  f  =5  times  YV=fl- 


[Give  principle  and  statement  as  in  third 
form.] 
Forth/arm  (   f  of  f  =4  of  5  times  |,  or  -1/. 

of    = 


70 


Written  f  win. 


(1.)     1X4=3. 

(2.) 

(3.) 

(4.) 


Fifth  form 


/.  etc. 

Thought  form. 

The  multiplicand  multiplied  by  its  denom- 
inator equals  its  numerator. 
<,  The  multiplier  multiplied  by  its  denomin- 
ator equals  its  numerator.  Multiplying  to- 
gether the  corresponding  members  of  these 
equations  gives  equation  (3)  ;  and  dividing 
both  members  of  equation  (3)  by  28  gives 
equation  (4). 

The  first  member  of  equation  (4)  consists 
of  the  two  factors  whose  product  is  required, 
while   the   second   member   is   the  product 
\sought. 


153.     General  Remarks  on  Multiplication   of  Fractions, 

1  Mixed  numbers  may  be  reduced  to  fractions  before  multi- 
plying; or  the  multiplicand  may  be  multiplied  by  the  integral 
and  the  fractional  parts  of  the  multiplier  (used  separately)  and 
the  partial  products  combined. 

2.  If  one  or  both  factors  be  decimal  fractions  expressed  in 
the  decimal  notation,  any  of  the  given  forms  may  be  used. 
The  second  form  is,  perhaps,  the  best  to  use  in  the  multiplica- 
tion of  decimal  tractions.  • 

Example.     Multiply  2.5  by  .025. 

Form.     The  multiplier  equals  .001  of  25. 

2.5X25=62.5. 

2.5  X.  001  of  25=.001  of  62.5=.0625. 

-.  2.5X.025=.0625. 


71 

3.  The  product  of  one  fraction  by  another  may  be  obtained 
by  multiplying  together  the  numerators  of  the  factors  for  the 
numerator  of  the  product,  and  the  denominators  of  the  fac- 
tors for  the  denominator  of  the  product.  [See  rule  in  any  good 
text  book  ] 

Exercises. 

Multiplicands,  f  j  8 ;  &  ;  3| ;  f  ;  £  j  ^  ;  f ;  f ;  21  . 
16;  12;  .05;. 3;  . 025;  3.2;  21.004;  8 J ;  4.07;  t;  .09  j 
.5  ;  .008 ;  3  J ;  41 ;  6.7 ;  8.09 ;  6.3  ;  32.04  ;  2.004  ;  325.046  j 
4.0064 ;  31.75. 

Multipliers,  f ;  f;  t;  £ ;  f ;  f;  f ;  j.  2.5  >5; 
.3;.7;  .04;  .005;  .045;  .008;  £;  1.5;^;  2.3;  4.05; 
6.25  ;  15.5  ;  7.5  ;  .47  ;  31.056  ;  .0045. 


Division  of  Fractions. 

154.     CASE  I. 

To  divide  by  an  integer. 

Example  1.     Divide  |-  by  3. 

Thought  form.  Since  a  fraction  is  divided  by  di- 
viding its  numerator,  -f-i-3=-f-. 

Example  2.  Divide  ^  by  5. 

Thought  form.  Since  a  fraction  is  divided  by  mul- 
tiplying its  denominator,  £-r-5—-f^. 

Exercises. 
Dividends.    l;f;|;    f;    f;:-|j    f;    #;«;«; 

H;  ft;  H;  -2;  -25; 3-25;  2-004;  36-78;  47-3;  17-08' 

6.25;  .0625  ;  3$ ;  5| ;  24|;  14}  ;   16f ;  26J;  32|. 

Divisors.     2,  3,  4,  5,  63  7,  8,  9,  10,  11,  12,  13,  14,  15. 


72 


155.     CASK  II.  t 

To  divide  by  a  fraction. 

Example  1.     Divide  5  by  f . 

f  5--l=5. 
5-1-4=7  times  5=35. 


Sec1  nd  form 


Third  form. 


f  The  divisor  equals  |  of  3. 


1  5-^1  of  3=7  times  f— ^=llf. 


--rr=5  times  i=4A =llf . 


Example  2.     Divide  f  by  f . 


Remark.     For  the  first  three  forms  see  those  given  for  the 
preceding  example. 

Fourth  form.  \  f — f-J-. 


Fifth  form. 


One  fraction  may  be  divided  by  another 
by  dividing  the  numerator  of  the  dividend 
by  the  numerator  of  the  divisor  and  the  de- 
nominator of  the  dividend  by  the  denom- 
inator of  the  divisor. 

If  the  terms  of  the  dividend  be  not  re- 
spectively divisible  by  the  corresponding 
terms  of  the  divisor,  they  may  be  made  so 
by  multiplying  both  terms  of  the  dividend 
by  the  product  of  the  terms  of  the  divisor. 

Division  of  fractions  is  thus  shown  to  be 
\the  reverse  of  multiplication  of  fractions. 


73 

156.     General  Remarks  on  Division  of  Fractions. 

1.  If  the  divisor  is  a  mixed  number,  it  is,  perhaps,  better  to 
reduce  it  to  a  fraction  before  dividing. 

2<  If  the  divisor  is  written  wholly  in  the  decimal  notation, 
the  above  forms  are  as  applicable  as  though  it  were  written  in 
the  fractional  notation.  The  second  form  is  preferred. 

Example.     Divide  .0625  by  2.5. 
Form.     The  divisor  is  .1  of  25. 

.0625-f-25=:.0025. 

.0625-f-.l  of  25=10  times  .0025^.025. 
.-.  ,0625--2.5=:.025. 

Exercises. 

Dividends.—  f  ;  f;  I;  $  ;  f  ;  f  ;  f  ;  .5  ;  .25  ;  .025  ;  8.4  ; 
3.2  ;  8.08  ;  10J  ;  3fr  ;  5*  ;  4f  ;  6*  ;  34.567  ;  325.568  ;  354.56  ; 
1.2342;  56.67895. 


.7  j  .5  ;  .3  ;  .07  ;  .002  j  .626  ;  1,256  ;  3.4467  ;  2£  ;  3i  ;  4f  ;  5|  } 


157.     Exercises  Involving  Fractions. 

1.  2  bu.=rwhat  part  of  3  bu.? 

2.  $4=  what  part  of  $10? 

3.  15  apples  =  what  part  of  50  apples  ? 

4.  ^  acre  =  what  part  of  4  acres  ? 

5.  f  of  a  gallon  ='what  part  of  5  gallons  ? 
6-  f  of  a  bu.=  what  part  of  10  bu  ? 

7.  f  of  an  orange  =  what  part  of  f  of  an  orange  ? 

8.  f==  what  part  of  f? 

9.  f=  what  part  of  I? 


74 

10.  $  =  what  part  of  |  ? 

11.  ^=  what  part  of  £  ? 

12.  f=what  part  of  £? 

13.  How  much  water  will   fill   4   tubs   if  each    tul) 
holds  5J  gallons? 

14.  What  cost  9  apples  at  1  \?  each  ? 

15.  If  81b.  of  sugar  sell  for  $1,  what   is  the  price  per 
pound? 

16.  At  $6  per  cord,  required   the  cost  of  I  ot  a  cord 
of  wood. 

17.  Kequircd  the  cost  of  -f-  Ib.  of  peaches   at  35/  per 
pound. 

18.  If  coal  is  $3  per  ton,  required  the  cost  of  4-f  tons. 

19.  If  a  train  run  15   miles   per  hr.,    how  far  will  it 
run  in  3J  hours? 

20.  At  $|  per   bushel,   required   the  cost  of  £  of  a 
bushel  of  corn. 

21.  What  cost  f   of  a  gallon   of  syrup,   at   %\\  per 
gallon  ? 

22.  If  a  man  cut  f  of  a  cord  of  wood   in  a  day,  how 
much  can  ho  cut  in  f  of  a  day  ? 

23.  A  boy  divided  ^  of  a  bushel  of  apples  among  4 
playmates;  what  part  of  a  bushel  did  each  receive? 

24.  A  man  had  i  of  a  barrel  of  pork  and  sold  f  of  it; 
what  part  of  a  barrel  remained  ? 

25.  A  man  lost£  of  his  money  and   found  \  us   much 
:IH  he  had  after  his  loss;  what  part  of  his   original  sum 
had  ho  then  ? 

26.^If  1  yard4of  ribbon   cost   $^,    how    muny  yards 
can  bo  bought  tor 


75 

t 

27.  How  many  yards  of  cloth    will  $10   buy  at  $2-i 
por  yard  ? 

28.  If  2  men  can  do  a  piece  of  work  in  4|-  days,  how 
long  will  it  take  8  men  to  do  it  ? 

29.  What  cost  3J  boxes  of  oranges,  if  2£  boxes  cost 
$9? 

30.  What  cost  30  bushels  of  corn,  if  3i  bushels  cost 
$1.20? 

31.  If  a  bushel  of  wheat  cost  $£,   what  cost  f  of  a 
bushel  ? 

4.2.  If  a  bushel  of  wheat  cost  $£ ,   how   many  bushels 
can  be  bought  for  $5.20  ? 

33.  A  girl  divided  10  apples   among  her  companions, 
giving  to  each  f  of  an  apple  ;  how  many  companions 
had  she? 

34.  4=f  of  what  number? 

35.  ^  of  14=f  Of  what  numbcr? 

36.  f  of  36=£  of  what  number  ? 

47.  I  of  $40=^£  of  the  cost   of  a   horse ;   required  its 
cost? 

37.  After  spending^  of  his  money,  John  had  $42  re- 
maining ;  how  much  had  he  at  first  ? 

39.  Iff  of  an  acre  of  land  be  worth  $15,  what  are  12 
acres  worth  ? 

40.  A  boy  sold  lemons  at  the   rate   of  6  for  8  cents  ; 
how  much  did  he  receive  for  3  lemons?  For  8  lemons  ? 
For  12  lemons? 

41.  If  f  of  a  yard   of  silk   cost  $3i ;   what   cost  4J 
yards  ? 

42.  If  §  of  a  yard  of  cloth  cost  $£,   what  cost  f  of  a 
yard? 


76 

43.  A  man   gained  $15   by   selling   a   watch   for  1^ 
times  its  cost ;  required  its  cost. 

44.  Mary,  after  losing  i  of  her  flowers,  had  but  3  re- 
maining; how  many  had  she  at  first? 

45.  If  to  $  the  cost  of  John's  coat  $10  be   added,  the 
sum  will  be  $21 ;  required  its  cost. 

46.  If  to  |  of  William's   age   8  years   be   added,  the 
sum  will  be  1^  times  his  age  ;  how  old  is  he  ? 

47.  Two  men  hire  a  wagon  for  $9  ;   A   uses  it  7  days, 
and  B  uses  it  2  days;  what  should  each  pay? 

48.  John  and  James  bought  22   apples  for  11  cents  ; 
John  paid  7  cents  while  James  paid  4  cents ;  how  many 
apples  should  each  receive  ? 

49.  Anna  has  5  pinks  more  than  Ruth,  and  together 
they  have  19 ;  how  many  has  each  ? 

50.  4  boy  said  that  4  is  3  less  than  \  his   number  of 
marbles  ;  how  many  has  he? 

51.  10  years  are  6  years  more  than  f   of  John's  age  ; 
how  old  is  he  ? 

52.  If  A  and  B  do  ^  of  a  piece   work   in  a  day,  how 
long  will  it  take  them  to  do  the  entire  work  ? 

53.  George  can  plow  a  field   in    8   days,    and  Henry 
can  plow  it  in  12  days ;  how    long   would  it  take  them 
to  do  the  work,  working  together  ? 

54.  If  %  of  a  barrel  of  flour  cost  $4f ,  what  cost  £  of  a 
barrel? 

55.  If  A  can  do  f  of  a  piece  of  work   in   a  day,  how 
much  can  he  do  i»  2  days  ? 

56.  What  cost  6  bushels  of  clover   seed,   if  2  bu.  cost 
$12f? 


77 

57.  If  .3  of  a  pound  of  coffee  cost  9/,  what  cost  f  of 
a  pound  ? 

58.*  A  has  $13  which  is  f  of  twice  as  much  as  B  has  . 
how  much  has  B  ? 

59.  A  horse  is  sold  for  $60  which   is  f  of  £  of  its  val- 
ue; required  its  value. 

60.  Henry  and  George  bought  30  nuts;   Henry  paid 
18  cents  and  George  paid  12  cents;  how  should  the  nuts 
be  divided? 

61.  A  man  failing  in  business  can  pay  40  cents  on  the 
dollar ;  what  part  of  his  debts  can  he  pay? 

62.  i=f  of  what  number  ? 

63.  A  has  $3  more  than  B  ;  and   both  together  they 
have  $71;  how  many  dollars  has  each? 

64.  £— f  of  twice  what  number? 

65.  If  f  of  a  box  of  berries  cost  $f-,  what  cost  I  of  a 
box? 

66.  What  is  the   number   if  its   i   increased  by  10 
equals  21  ? 

67.  Required    the  number  if  its    £   added   to   its    £ 
equals  f. 

68.  f  of  a  number  -(-5=26  ;  what  is  the  number? 

69.  A  farmer  sold  f  of  his  grain,  and' had  120  bushels 
remaining ;  how  much  had  he  at  first  ? 

70.  Required  the  cost  of  15  horses  at  the  rate  of  3} 
horses  for  $169. 

71.  How  much  will  3J  acres  of  land   cost  at  $64  for 
H  acres? 

72.  The  difference  between  f  of  a  number  and  £  of  it 
is  6;  what  is  the  number? 


78 

73.  If  2J  times  a  number  exceed  2  times  the  number 
by  3f ,  what  is  the  number  ? 

74.  If  a  man  walk  |-  of  a   mile  in    lOf  minutes,  how 
long  will  it  take  him  to  walk  5  miles  ? 

75.  Eequired  the  cost  of  4  dozen  eggs  at  25  cents  for 
10  eggs. 

76.  Eequired  the  cost  of  45   apples   at   10  cents  per 
dozen. 

77.  $6=|  of  i  of  a  sum  of  money  ;  required  the  sum. 

78.  How  many  yards  of  goods   will    make  3  dresses 
if  15  yards  make  f  of  a  dress  ? 

79.  |  of  the  length  of  a  pole  is   in  the  water  and  15J 
feet  are  out ;  what  is  the  length  of  the  pole  ? 

80.  If  19  boxes  of  berries  are  worth  57  cents,  requir- 
ed the  value  of  f  of  a  box. 

81.  If  f  of  a  yd.  of  cloth  cost  $lf,  what  cost  $>f  yd.? 

82.  f  of  John's  money  equals   ^   of  Harry's,  and  to- 
gether they  have  $55  ;  how  much  has  each  ? 

83    If  on  1  orange  I  lose   ^   of  a   cent,    h^w    many 
oranges  must  I  sell  to  lose  6  cents  ? 

84.  William  has  twice  as  many  cents  as  Herbert,  and 
together  they  have  24  ;  how  many  has  each  ? 

85.  Two  boys  have  49  marbles ;  one  has  7  more  than 
the  other;  how  many  has  each  ? 

86.  Henry  received  for  his  horse  -J  of  its  cost ;   what 
part  of  the  cost  was  the  gain  ? 

87.  A  coat  which   cost   $12   was   sold   for  $16;  the 
gain  was  how  many  hundredths  of  the  cost  ? 

88.  Goods  bought   at   $12   were   sold   for  $10 ;  how 
many  hundredths  of  the  cost  was  the  loss? 


79 

89.  A  boy  bought  some  apples  for  72  cents  and  sold 
them  for  84  cents  ;  the  gain  was  what  part  of  the  cost? 
How  many  hundredths  of  the  cost? 

90.  A  horse  was  bought  for   $60   and   sold   for  $48  ; 
what  part  of  the  cost  was  the  loss  ?      How  many  hun- 
dredths of  the  cost  was  the  loss  ? 

91.  f  of  $6^how  many  hundredths  of  $20. 

92.  For  what  must  goods  costing  $50  be  sold  to  gain 
ten  hundredths  of  the  cost  ? 

93.  A  paid  $80  for  a  horse   and   sold   it   so  as  to  lose 
Of  its  cost    for  what  did  he  sell  it  ? 


94.  If  a  merchant   sold   goods   at  $2   per  yard  and 
thereby  gained  ^5^  of  the  cost,  required  the  cost. 

95.  A  cow  was  bought  for  $25  and  sold  for  $30;  what 
part  of  the  cost  was  the  gain  ? 

96.  Henry  can  make  f  of  a  pair  of  boots   in  a  day, 
and  James  can  make  %  of  a   pair  in  a  day  ;  how  long 
will  it  take  both  to  make  2  pairs  of  boots  ? 

97.  In  how  many   days  can   3   men  cut  15  cords  of 
wood,  if  1  man  in  1  day  cut  f  of  a  cord  ? 

98.  A  boy  bought   a  certain   number  of  apples  at  2 
cents  each,  and  the  same  number   at  4  cents  each,  and 
then  sold    out    at    the  rate  of  3  for  5   cents  ;  did  he 
gain  or  lose  and  how  much  ? 

99.  Iff  of  an  orange  cost  as  much  as  I  of  a  pineap- 
ple, required  the  price  of  two  oranges  in  pineapples. 

100.  A  man   gained  $10   by  selling   hia  horse   for  If 
times  its  cost;  what  was  the  cost  ? 

101.  A  can  do  a  piece  of  work  in  5  days  and  B  can  do 
it  in  3  days  ;  in  what  time  can  they  together  do  it  ? 


80 

102.  M.  bought f of 15J  yd.    of  cloth   for   f  of  $241; 
required  the  price  of  the  cloth  per  yd. 

103.  If  95  bushels  of  apples  cost   $110;  what   is   the 
value  of  3i  bushels.  ? 

104.  A  lumber  dealer  bought  siding   at  $18.75  per  M 
and  sold  it  at  $2.875  per  C ;  how  much  did  he  gain  per 
M? 

Aliquot  Parts. 

158.  Table  of  Aliquots. 

2|  =  J  of    10.  18!  =  A  of  100. 

3£  =  J  of    10.  20    =  i  of  100. 

6J  =3^  of  100.  .         25   =  J  of  100. 

gJ^J^of  100.  33£  =  £of  100. 

12^  =  |  of  100.  62J  —  |  of  100. 

16§  =  J  of  100.  66f  ==  |  of  100. 
125  =  i  of  1000. 

159.  Forms  of  Solution. 

Example  1.  At  181  cents  per  Ib.  required  the  cost 
of  32  Ib.  of  butter. 

Solution.  At  $1  per  Ib.  32  Ib.  of  butter  cost  $32,  but 
at  18!  cents,  or  $^,  32  Ib.  cost  T\  of  $32,  or  $6.  .-.  etc, 

Example  2.  At  12^  cents  per  Ib.  how  many  Ib.  of 
rice  will  $24  buy? 

Solution.  At  12£  cents,  or  $£  per  Ib.,  $1  will  buy  8 
Ib.,  and  $24  will  buy  24  times  8  Ib.,  or  192  Ib.  .-.  etc. 

Example  3.  At  $87?  per  acre,  how  many  acres  of 
land  can  be  bought  for  $4900? 

Solution,  At  $100  per  acre,  $4900  will  buy  49 
acres,  but  $87  J,  or  J  of  $100,  $4900  will  buy  8  times 
of  49  acres,  or  56  acres. 


81 

160.     Exercises. 

1.  At  6i/  each  what  cost  16  oranges  ? 

2.  At  $6.25  per  barrel  what  cost  7  barrels  of  ftour? 

3.  At  2JX  apiece  what  cost  20  pencils? 

4.  At  $2.50  a  box  what  cost  60  boxes  of  potatoes  ? 

5.  At  8JX  per  yd.  what  cost  56  yd.  of  muslin  ? 

6.  At  $8.33£  each  what  cost  30  calves? 

7.  At  12£X  a  lb.  what  cost  64  Ib.  of  sugar? 

8.  $12.50  per  acre  what  cost  17  acres  of  corn  ? 

9.  At  16fX  per  lb.  what  cost  40  lb.  of  butter? 

10.  At  16fX  each  what  cost  4  slates  ? 

11.  At  18t/  per  lb.  what  cost  32  lb.  of  steak? 

12.  At  $18.75  each  what  cost  80  ponies  ? 

13.  If  1  doz.  eggs  cost  20/  what  cost  15  doz.  ? 

14.  At  20/  each  what  cost  7  books.  ? 

15.  If  1  cow  cost  $25,  required  the  cost  of  16  cows. 

16.  At  25/  each  what  cost  28  collars  ? 

17.  At  3^/  apiece  what  cost  9  apples  ? 

18.  At  $3.50  each  what  cost  15  hats  ? 

19.  At  $33£  per  acre  what  cost  30  acres  of  land? 

20.  At  50/  each  what  cost  7  books  ? 

21.  At  62|/  each  what  cost  17  pitchers  ? 

22.  At  $661  per  head  what  cost  12  horses? 

23.,  At  $1.25  per  rod  what  cost  10  rods  of  fencing? 

24.  At  $125  peivhead  what  cost  14  horses.  ? 

25.  At  $20  per  acre  what  cost  15  acres  of  land  ? 

26.  At  37£/  per  yd.  what  cost  7  yd.  silk  cord  ? 

-27.  At  $75  per  head  required  the  cost  of  12  horses. 
28.  At  6J/  per  spool  required  the  cost  of  11  spools 
of  thread. 


82 


SECTION  VIII. 
COMPOUND  NUMBERS. 

161.  Compound  numbers  are  classified  on  the  basis 
of  the  kind  of  attribute  measured,  as — 

1.  MEASURE  OF  DURATION. 

2.  MEASURES  OF  EXTENSION. 

3.  MEASURES  OF  FORCE. 

162.  Diagram  exhibiting  in  classified  form  the  "meas- 
ures" usually  treated  under  compound  numbers. 

Remarh.  1.  The  decimal  measures  are  embraced  in  the  dia- 
gram ,  but  are  treated  separately. 

2.  The  measure  of  value  in  most  civilized  countries  is  derived 
from  the  force  of  gravity,  or  weight.  The  primary  units  of 
value  were  weight  units. 

1.  Of  Duration. — Time  measure. 


f  Long  measure. 
I  Surveyors'  long  measure, 
Length  -{  Mariners'       " 
|  Decimal 
t  Circumference 


2.  Of  Extension 


Measures. 


3.  Of  Force 


(  Square  measure. 
Surfaced  Surveyor's  square    " 
(  Decimal 

f  Cubic  measure. 
I  Decimal  cubic  measure. 
Volume  \  Dry  measure. 

|  Liquid  measure. 
I  Decimal  capacity 


{Troy  Weight. 
Apothecaries'  weight. 
Avoirdupois 
Decimal 

{United  States  money. 
English 
Etc. 


163.  Order  of  Study. 

Remark.    The   following  order  should  be  followed  in  the 
study  of  each  measure. 

(1.)  The   primary,   or  standard  unit.     How   deter- 
mined. 

(2.)  Other  units  and  their  relative  value. 
(3.)  Scale  and  table. 
(4  )  Reduction. 
a.  Descending, 

(  1.  Integers  to* integers  of  lower  denomination. 
1  2.  Fractions "  "       •'  " 

(3.  « fractions'-       " 

b.  Ascending. 

C  1.  Integers  to  integers  of  higher  denomination. 

X  2.         "         "  fractions "       "  <•         <; 

( 3.  Fractions"        "        a      u  u 

(5.)  Synthesis. 

a.  Addition. 

b.  Multiplication. 
(6.)  Analysis. 

a.  Subtraction. 

b.  Division. 

164.  Tables. 

For  the  tables  of  compound  numbers  and  many 
interesting  and  useful  facts  the  pupil  is  referred  to 
textbooks  on  Arithmetic  and  to  Encyclopedias. 


84 

Applications  of  Compound  Numbers. 

Remark.  The  "order  of  study"  will  be  applied,  in  part,  to 
"time  measure."  Each  of  the  other  measures  should  be  treat- 
ed in  a  similiar  manner. 

*  165.     Time  Measure. 

Time.  That  which  renders  succession  possible  is 
called  time. 

The  Primary  Unit.  The  average  solar  day  is 
taken  as  the  primary  unit  of  time. 

(1.)  A  Solar  day  is  the  interval  of  time  between 
two  successive  transits  of  the  vertical  rays  of  the  sun 
across  a  given  meridian.  This  interval  varies  at  dif- 
ferent times. 

(2.)  The  sidereal  day  is  the  interval  between  two 
successive  transits  of  a  fixed  star  across  a  given  merid- 
ian. This  interval  is  the  same  at  one  time  that  it  is  at 
another. 

(3.)  The  solar  and  the  sidereal  days  would  be  of 
equal  length  if  the  earth  did  not  revolve  around  the 
sun.  While  the  earth  is  rotating  upon  its  axis  it  is 
also  moving  forward  in  its  orbit ;  so  that  when  it  has 
made  a  complete  rotation,  it  must  make  part  of  an- 
other before  the  sun's  rays  are  vertical  a  second  time 
upon  any  given  meridian.  The  solar  day  is  thus  a 
little  longer  than  the  sidereal  day.  [About  four  min- 
utes.] 

Other  Denominations.  The  other  time  units  are 
either  multiples  or  divisors  of  the  day. 

The  multiples  of  the  day  are  the  week,  the  month, 
the  year  and  the  century. 

The  divisors  of  the  day  are  the  hour,  the  minute 
and  the  second. 


85 

Relations. 

Multiples. 

1.  The  week  equals  7  days. 

2.  The  month  equals  4|-  weeks. 

Remark.  The  average  calendar  month  is  a  little  more  than4^ 
weeks,  or  30  days,  while  the  lunar  month  is  a  small  fraction 
more  than  4  weeks. 

3.  The  year  equals  12  calendar  months. 

4.  The  century  equals  100  years. 

Divisors, 

1.  The  hour  equals  •£%  of  the  day. 

2.  The  minute  equals  -^  of  the  hour. 

3.  The  second  equals  ^  of  the  minute. 

Scale.  The   units   used  in   time  measure   may  be 
written  in  a  scale  as  follows: 

100          12         4f  7         24         60          60 

cen.       yr.       mo.       wk.       da.      hr.      min.      sec. 

11111111 
Remarks.    1.  The  values  expressed  by  the  respective  units 
of  this  scale  increase  from  right  to  left  in  the  written  scale.     In 
this  respect  the  time  scale  is  like  the  decimal  scale. 

2.  Since  the  rate  of  increase  varies,  the  time  scale  is  called  a 
varying  scale.     In  this  respect  it  is  unlike  the  decimal  scale, 
whose  rate  of  increase  is  uniformly  10. 

3.  The  time  scale  consists  of  but  eight  units.     In  this  re- 
spect it  is  unlike  the  decimal  scale,   whose  orders  of  units  may 
be  repeated  in  periods  indefinitely. 

4.  In  the  time  scale  the  units  extend   both  above  and  below 
the  primary  unit.    In  this  respect  it  is  like  the  decimal  scale. 

Table.     For  convenience  the  relations  of  the  time 
units  may  be  tabulated  thus  : 

60     sec.    =  1   min. 
60     min.  =  1  hr. 
24     hr.      =  1  da. 
7     da.      =1  wk. 
4f  wk.    =  1   mo. 
12     mo.    =  1  yr. 
100    yr.     =  1  cen. 


Reduction. 
166.     Reduction  Descending. 

Example  1.     Eeduce  2  wk.    3  da.    12   hr.    to  min. 

First  form,  v  1  wk.  =  7  da., 

2    "      ==  2  times  7  da.  =  14  da. 
14  da. +3  da.    =  17  da. 
v  1  da.  =  24  hr., 

17    "     =  17  times  24  hr.  =  408  hr. 
408  hr.-f  12  hr.    :  =  420  hr. 
V    1    "     =  60  min., 

420   "     =  420  times  60  min.  =  25200  min. 
.-.2  wk.  3  da.  12   «     ==  25200  min. 

Second  form.  v  1  wk.  =  7  times  1  da., 

2  *<  =  7      "      2   "  =14  da. 

14  da.-f3  da.  =  17  da. 

V  1    "  =24  times  1  hr., 

17  "  =  24   "  17  "  ==  408  hr. 

408  hr.-f  12  hr.  =  420  hr. 

•.•  1  "  =  60  times  1  min., 

420  "  =±  60      «     420  "   =  25200  min. 

.-.  2  wk.  3  da.   12   «  =  25200  min. 

Example  2.     Koduce  f  wk.  to   smaller  denominate 
units. 

First  form,     v  1  wk,  — .  7  da., 

|   a     =|of7da.  =  3|  da. 

v  1  da.   ==  24  hr., 

I    "       =  f  of  24  hr.  =  21J  hr. 

•/I  hr.    =  60  min.,,, 
J   "     =  J  of  60  min  —  20  min. 

/.  |  wk.  —  3  da.  21  hr.  20  min. 


87 

• 

Second  form,  v  1  wk.  =  7  times  1  da., 

f   "     =7  times  f  da.  =  3f  da. 

v  1  da.  =  24     "     1  hr., 

f   «     =24     «     I  "   =  21J  hr. 

V  1  hr.   ^=  60     "     1  min., 

J   "    =  60    "    J    «    =  20  min. 

.-.  |  wk.  =  3  da.  21  hr.  20  min. 


Example  3.     .Reduce  ^^  of  a  wk.  to  the  fraction  of 
an  hour. 

First  form,     v  1  wk.  =  7  da., 

vfa"    =  Tfa  of  7  da.  =  .03  da. 
V  1  da.  =  24  hr., 
.03   "    =  .03  of  24  hr.  =  £f  hr. 
wk.  =       hr. 


Second  form,  v  1  wk.  =  7  times  1  da., 

Tfr  "    =7    "    T*TF"  =-03  da. 

v  1  da.  =  24   "      1  hr., 
.03  "     =  24   «     .03  "   =  -Jf  hr. 
wk.  ==  X    hr. 


(1.)  Two  forms  of  solution  have  been  given  for  a 
problem  in  each  case  under  reduction  descending.  The 
first  form  in  each  case  is  that  usually  given  for  such 
problems  and  needs  no  comment  other  than  the  ob- 
servation that  the  multiplier  is  not  taken  from  the 
table  but  is  the  number  (taken  abstractly)  to  be  re- 
duced. 

(2.)  The  second  form  rests  upon  the  following  : 

PRINCIPLE.  —  The  numerical  relation  that  exists  be- 
tween given  units  exists  between  like  multiples  and  also  be- 
tween like  parts  of  those  units. 

[See  remark  under  Prin.  VII,  page  58.] 


The  first  step  consists  in  the  statement  of  the 
relation  existing  between  a  unit  of  the  denomination 
to  be  reduced  and  a  unit  of  the  denomination  to  which 
the  reduction  is  to  be  made.  The  second  step  is  made 
in  the  light  of  the  above  principle,  e.  g.  Since  1  hr. 
=60  times  1  min.,  420  hr.  (a  multiple  of  1  hr.)  equal 
60  times  420  min.  (a  like  multiple  of  1  min.)  It  is  ob- 
served that  the  multiplier  is,  in  every  instance,  taken 
from  the  table.  This  form  of  solution  is  uniform  and 
general  in  its  application  to  the  solution  of  all  problems 
in  the  several  cases  of  reduction  descending  in  all  the 
measures. 

(3.)  A  careful  study  of  the  second  form  given  for 
the  solution  of  problems  in  reduction  ascending,  will 
show  the  form  to  be  uniform  and  of  general  application 
in  all  the  measures. 

(4.)  Any  reduction,  either  descending  or  ascend- 
ing, may  be  made  by  a  direct  use  of  the  equation.  The 
first  statement  under  the  second  form  in  each  of  the 
given  examples  is  taken  as  the  first  equation.  The 
second  equation  is  obtained  from  the  first  by  multiply- 
ing both  its  members  by  such  a  number  as  will  give 
the  number  to  be  reduced  for  the  first  member  of  the 
second  equation.  In  transforming  the  first  equation  it 
is  to  be  observed  that  its  second  member  consists  of 
two  factors,  and~that  the  member  is  to  be  multiplied  by 
multiplying  its  second  facfor.  [Prin.  X,  page  31.] 

167.     Reduction  Ascending. 

Example  1.  Reduce  25200  min.  to  integers  of  high- 
er denomination. 


First  form,     v  60  min.  =  1  hr., 

25200     ll   =  as  many  hr.  as  25200  min. 

are  times  60  min.  which  =  420. 
V  24  hr.  =  1  da., 

420  hr.  =  as  many  da.  as   420  hr.  are 
times  24  hr.  which    =  17.    with  a  re- 
mainder of  12  hr. 
v  7  da.  =  1  wk., . 

17  da.  =  as  many  wk.  as  17  da.  are 
times  7  da.  which  =  2,   with   a  re- 
mainder of  3  da. 
.-.  25200  min.  =  2  wk.  3  da.  12  hr. 

Second  form,     v  1  min.  =  -fa  of  1  hr., 

25200     "    ==  gV  of  25200  hr.  =  420  hr. 
v  1  hr.    =  fa  of  1  da., 
420   "      =  fa  of  420  da.  =  17  da.  12  hr. 
V  1  da.    =  |  of  1  wk., 

17   «     =  \  of  17  wk.  ==  2  wk.  3  da. 
.-.  25200  min.  =  2  wk.  3  da.  12  hr. 

Example  2.  Eeduce  3  da.  21  hr.  20  min.  to  the  frac- 
tion of  a  wk. 
First  form.     v60  min.  =  1  hr., 

20  u     —  as   many   hr.   as   20  min.  are 
times  60  min.  which  =  £. 

21  hr.  -f  J  hr.  =  21J  hr. 
-.•24  hr.  =  1  da., 

24£  hr.  =  as  many  da.   as   21J   hr.   are 

times  24  hr.  which  =  %. 
3  da.  +  f  da.  ==  3|  da. 
v  7  da.  =  1  wk., 

3|  da.  —  as  many  wk.   as   3f  da.   are 

times  7  da.  which  =  ^. 
.-.  3  da.  21  hr.  20  min.  =  f  wk. 


90 

Remark.  The  foregoing  form  is  the  same  as  the  first  form 
used  in  solving  Ex  1.  The  following  form  is  more  in  accord- 
ance with  the  facts  involved. 

v  60  min.  =  1  hr., 

20  "     =  such  part  of  1  hr.  as  20  is  part   of  60 
which  is  J. 

21  hr.  +  J  hr.  =  21 J  hr. 
v  24  hr.  —  1  da.,. 

21J  hr.  =  such  part  of  1  da.  as  21^  is  part  of  24 
which  is  f . 

3  da.  +  f  da.  =  3f  da. 
v  7  da.  =  1  wk., 

3f  da.  =  such  part  of  1  wk.  as   3f   is  part   of  7 
which  is  -|.     /.  etc. 

Second  form,     v  1  min.  =  ^  of  1  hr., 

20  "     =  fa  of  20  hr.  ==  $  hr. 

21  hr.  -f  i  hr.  =  21J  hr.  =  *£.  hr. 

v    1  hr.  =  Jj  of  1  da., 

.ayt  br.  =  J¥  of  %4-  da.  =  f  da. 

3  da.  -f  |  da.  =  3|  da.  =  -395-  da. 
v    1  da.  •=  -J-  of  1  wk., 

y  da.  =  i  of-3/  wk,  =  |  wk. 
.  .-.  3  da.  21  hr.  20  min.  =  |  wk. 

Remark.  The  several  denominate  numbers  constituting  the 
given  compound  number,  may  be  reduced  to  the  lowest  de- 
nomination (min.)  ;  and  this  number  of  minutes  may  then  be 
reduced  to  the  fraction  of  a  week.  Mixed  numbers  are  thus 
avoided. 

Example  3.  Eeduce  if  hr.  to  the  fraction  of  a  wk. 

First  form.  [Use  either  the  first  form  given  under 
example  2  or  that  given  in  the  remark  under  exam- 
pie  2.] 


91 

Second  form,     v    1  hr.  =  -^  of  1  da., 

if  hr.=  jfcofjf  da.  =  .03  da. 
v    1  da.  _  \  of  1  wk., 

.03  da.  —  |  of  .03  wk.  =  ^ T  wk. 
/.  if  hr.  =  T^  wk. 

168.     General  Remarks.  *' 

1.  Such  examples  as  the  following  are  often  given. 

2  da.  5  hr.  15  min.  equal  what  part  of  2  wk.  4.  da. 
3  hr.  ? 

In  such  cases   each   compound   number  should  be 
reduced  to  the  lowest  denomination  given  in  either. 
Solution. 

2  da.  5  hr.  15  min.  ==  3195  min. 

2  wk.  4  da.  3  hr.  =  26100  mjn. 
v  1  min.  =  26;00  of  26100  min., 
3195  min,  =  3195   times    -^-^    of    26100   min.  = 

$&  of  26100  min. 
.-.  etc. 

2.  The  following  two  solutions  are  given  for  the  purpose  of 
exhibiting  a  method  that  shall  obviate  the  use  of  a  complex 
fraction  in  stating  the  relation  of  the  less  unit  to  ihe  greater  in 
each  example. 

Example  1.  Reduce  33  yd.  to  rd. 

The  first  step  (using  the   second  form  given  in  ex- 
amples 1,  2  and  3  in  Reduction   Ascending,)  is  to  state 
the  relation  between  1  yd.  and  1  rd.,  thus: 
V  1  yd.  =  -gij  of  1  rd.,  etc. 

Since  the  relation  as  stated  cannot  be  read  as  a 
fraction,  it  is  well  to  express  the  relation  by  -fa  which 
is  the  equivalent  of  -5\j. 

The  solution  should  be  as  follows : 


fv    ljd. 

J      33   "    = 
(.-.33    u    == 


Form.  4      33   "    =  T2T  of  33  rd.  =  6  rd. 


92 


Example  2.  Eeduce  132  ft.  to  rd. 

(  V 
i.  \ 
L-. 


1  ft.  =  ^  Of  1  rd., 

Form.  I      132  "  =  •&  of  132  rd.  =  8  rd. 
132  "   =  8  rd. 

3.  Addition,  multiplication,  subtraction  and  division  of  com- 
pound numbers  may  be  effected  in  the  same  manner  and  in 
obedience  to  the  same  principles  that  govern  the  synthesis  and 
the  analysis  of  numbers  expressed  in  the  decimal  scale.    No 
new  principle  is  introduced  and  but  one  new  fact,  viz.: — a  va- 
rying scale  is  employed  instead  of  a  uniform  scale. 

4.  The  subject  of  time  is  placed  first  in  the  classification  of 
the  measures  because  the  primary  units  of  extension  and 
weight  are  derived  from  a  time  unit. 

The  primary  unit  of  the  common  measures  of  extension  is 
the  yard,  which  is  a  definite  portion  of  the  length  of  a  pendu- 
lum that  beats  seconds  under  certain  conditions. 

The  primary  unit  of  weight  is  the  pound  which  is  the  force 
of  gravity  that  acts  upon  a  certain  volume  of  water  under  cer- 
tain conditions.  The  primary  unit  of  extension  is  derived  direct- 
ly from  a  time  unit  (the  second),  and  the  primary  unit  of  weight 
is  derived  directly  from  a  measure  of  extension  (a  volume) 
arid  through  that  from  the  same  time  unit,  (the  second.) 

The  primary  unit  of  value  is  a  certain  weight  of  silver  or 
gold. 

[NOTE— Circumference  measure  and  the  decimal  (metric)  measures  are 
exceptions  to  the  above  remark.] 

169.     Exercises. 

1.  Reduce  5  wk.  3  da.  4  hr.  to  min. 

2.  Reduce  f  da.  T25  hr.  T7T  min.  to  sec. 

3.  Reduce  2  bu.  3  pk.  5  qt.  to  pt. 

4.  Reduce  I  bu.  f  pk.  to  qt. 

5.  Reduce  5  yd.  2  ft.  7  in.  to  in. 

6.  Reduce  2  mi.  to  fathoms. 

7.  Reduce  60  acres  to  sq.  ft. 

8.  Reduce  3  hhd.  24  gal.  3  qt.  to  gills. 

9.  Reduce  4  Ib.  15  pwt,  to  grains. 

10.  Reduce  £  oz.  f  pwt.  to  grains. 

11.  Reduce  7$  2£  to  grains. 


93 

12.  Eeduce  f  |  £3  §9  to  grains. 

13.  Eeduce  5  cwt.  14  Ib.  7  oz.  to  drams. 

14.  Reduce  j-  ton  f  cwt.  -§-  Ib.  -^  oz.  to  oz. 

15.  Reduce  7  wk.  3  da.  1  hr.  15  min.  to  seconds. 

16.  Reduce  3  gal.  2  qt.  1  gi.  to  gills. 

17.  Reduce  3  yd.  2  ft.  5  in.  to  inches. 

18.  Reduce  5  bu.  2  pk.  7  qt.  to  pints. 

19.  Reduce  f  Ib.  Troy,  to  integers  of  lower  denom- 
inations. 

20.  Reduce  .3  da.   to  integers  of  lower  denomina- 
tions. 

21.  Reduce  f  yd.  to  integers  of  lower  denomina- 
tions. 

22.  Reduce  .875  gal.  to  integers  of  lower  denomin- 
ations. 

23.  Reduce  y^Vs"  da.  to  the  denomination  minutes. 

24.  Reduce  .007  gal.  to  the  denomination  pints. 

25.  Reduce  -^  yd.  to  the  denomination  inches. 

26.  Reduce  -^  bu.  to  the  denomination  pints. 

27.  Reduce  54960  min.  to  integers  of  higher  de- 
nominations. 

28.  Reduce  186  pt.  to  a  compound  number  of  higher 
denominations. 

29.  Reduce  57648  sec.  to  a  compound  number  of 
higher  denominations. 

30.  Reduce   35J  qt.  to  a  compound  number  com- 
posed of  pk.  and  bu. 

31.  Reduce  211  inches  to  a  compound    number 
composed  of  in.  ft.  and  yd. 

32.  Reduce  23400  grains  to  Troy  integers  of  higher 
denominations. 


94 

33.  Reduce  600  gr.  to  a  compound  number  com- 
posed of  pwt.  and  ounces. 

34.  Reduce  8211  oz.  to  a  compound  number  com- 
posed of  Ib.  and  cwt. 

35.  Add  together   3  bu.   3  pk.  6  qt.  1  pt.;  5  bu.  2 
pk.  5  qt.  1  pt.;  6  bu.  1  pk.  7  qt.  1  pt.;  1  bu.  3  pk.  2  qt. 

36.  Add  34  cwt.  17  Ib.  11  oz.  13  dr.;  19  cwt.  46  Ib. 
7  oz.  4  dr.;  71  cwt.  10  Ib.  15  oz.  12  dr. 

37.  Add  5  mi.  6  fur.  21  rd.  4  yd.  2  ft.  9  in.;   9  mi. 
7  fur.  17  rd.  5  yd.  1  ft.  11  in,;  14  mi.  21  fur.  14  rd.  2  yd. 
2  ft.  4  in. 

38.  From  24  sq.  yd.  7  sq.  ft.  110  sq.  in.  take  16  sq. 
yd.  6  sq.  ft.  136  sq.  in.    . 

39.  From  63  bu.  2  pk.  5  qt.  take  54  bu.  3  pk.  6  qt. 

40.  From  a  hhd.  of  molasses  f  leaked  out.     How 
much  remained? 

41.  What  is  the  time   from  March  16,  1884  until 
July  4, 1884? 

42.  How  long  has  it  been  since  February  4,  1884  ? 

43.  A  note  was  given  Dec.  18,  1883  and  was  paid 
March  28, 1884;  how  long  did  it  run  ? 

44.  Multiply  5  Ib.  6  oz.  3  pwt.  11  gr.  by  6. 

45.  Multiply  4  bu.  3  pk.  7  qt.  1  pt.  by  5. 

46.  Multiply  4  rd.  3  yd.  2ft.  7  in.  by  4. 

47.  Divide  5  da.  13  hr.  44  min.  18  sec.  by  6. 

48.  Divide  6  cwt.  44  Ib.  12  oz.  10  dr.  by  4. 

49.  Divide  3  bu.  2  pk.  5  qt.  by  2  bu.  6  qt.  1  pt. 

50.  Divide  6  bl.  21  gal.  5  qt.  by  5  bl.  12  gal.  7  qt. 

51.  Reduce  f  of  aft.  to  the  fraction  of  a  rd. 

52.  Reduce  .05  of  a  mi.  to  lower  denominations. 

53.  -J  of  a  dr.  to  the  fraction  of  a  Ib. 


95 

SECTION  IX. 
TIME  AND  LONGITUDE. 

Distance  measured  on  a  parallel  of  latitude  is  called 
longitude. 

Remarks.  1.  Since  longitude  is  measured  on  the  arc  of  a 
circle  it  is  estimated  in  units  of  the  circumference,  viz.:  degrees, 
minutes  and  seconds. 

2.  Unless  otherwise  designated,  the  meridian  of  ^Greenwich 
is  taken  as  the  prime  from  which  longitude  is  reckoned. 

170.  Because  of  the  daily  rotation  of  the  earth  on 
its  axis,  any  place  on  the  surface  of  the  earth  except 
the  poles,  moves — 

In  24  hours  through  360°  of  space. 
"     1  hour        "  15°     « 

"     1  min.  15'      " 

"     1  sec.          «  15"    " 

Hence  if  15  units  of*  longitude  lie  between  two 
places,  the  time  registered  at  the  places,  respectively, 
differs  by  1  time  unit ;  hours  corresponding  to  degree?, 
minutes  to  minutes,  and  seconds  to  seconds. 

On  the  other  hand,  if  the  time  of  two  places  is  found 
to  differ  by  1  time  unit,  the  places  are  distant  from 
each  other  (east  and  west)  15  corresponding  longitude 
units. 

171.  All  places  on  the  earth's  surface  have  the  same 
absolute  time,  but  not  the  same  relative  time. 

When  the  vertical  rays  of  the  sun  reach  the  meridi- 
an of  a  place  it  is  noon  at  all  places  on  that  meridian  ;. 
while  it  is  afternoon,  or  later  in  the  day,  at  all  places 
east,  and  before  noon,  or  earlier  in  the  day,  at  all  places 
west  of  the  given  meridian.  Thus  a  place  east  of  a 
given  meridian  has  later  relative  time  and  a  place 
west  has  earlier  relative  time  than  that  on  the  given 
meridian. 


96 
172.     Standard  Time. 

For  the  purpose  of  estimating  time  for  the  railway 
service,  the  country  of  the  United  States  is  divided 
into  belts  or  strips  marked  by  meridians  of  longitude 
15°  apart.  The  75th  meridian  from  Greenwich  marks 
the  middle  of  the  eastern  belt.  The  90th  meridian 
marks  the  middle  of  the  central  belt.  The  105th  me- 
ridian marks  the  middle  of  the  mountain  belt.  The 
120th  meridian  marks  the  middle  of  the  Pacific  belt. 

The  local  time  on  each  of  these  meridians  is  taken 
as  the  "Standard"  time  for  railway  purposes  at  all 
places  within  the  given  belt;  and  since  these  meridi- 
ans are  15°  apart,  the  time  in  each  belt  is  registered 
one  hour  earlier  than  in  the  belt  next  at  the  east,  e  g. 
When  it  is  noon  on  the  75th  meridian,  it  is  noon  at  nil 
places  within  the  eastern  belt,  11  a.  m.  on  the  90th 
meridian  and  at  all  places  within  the  central  belt, 
10  a.  ra.  on  the  105th  meridian  and  at  all  places  within 
the  mountain  belt  and  9  a.  m.  on  the  ]20th  meridian 
and  at  all  places  within  the  Pacific  belt. 

Remark.  In  each  belt  near  its  prime  meridian  workshops, 
manufactories,  public  schools  and  many  other  branches  of 
business  are  now  generally  carried  on  by  "Standard"  instead 
of  local  time. 

• 

173.     Forms  of  Solution. 

Example  1.  The  time  between  two  places  is  6  hr. 
12  min.  10  sec.;  required  their  difference  in  longitude. 

fkrm. 

-.'  1  hr.  1  min.  1  sec.  correspond  to  15  times  1°  1'  1", 
6  hr.  12  min.  10  sec-  correspond  to  15  times  6°  12' 
10"=  93°  2'  30". 

.-.  their  difference  in  longitude  is  93°  2'  30". 


97 

• 

Example  2.  The  difference  in  longitude  between 
New  York  and  Greenwich  is  74°  3";  required  their  dif- 
ference in  time. 

Form. 

'.- 1°  V  1"  correspond  to  TV  of    1  hr.  1  min.  1  sec., 
74°  3"  "  "  TV  "   74  "    3sec.  =4hr.56 

min.  .2  sec.     .-,  etc. 

Example  3.     When  it  is  noon  at  New  York,  what 
is  Greenwich  time? 
Remark.    Their  time  difference  is  given  ab^ve 

Form.  Since  their  time  difference  is  4  hr.  56  min. 
.2  sec.,  and  since  Greenwich  is  east  of  New  York,  its 
time  is  later  than  that  of  New  York  by  4  hr.  56  min 
.2  sec.,  or 56  min.  .2  nee  past  4  p.  m. 

Example  4.   When  it  is  noon  at  Greenwich  what  is 
New  York  time? 
JSemark.    Their  time  difference  is  given  above. 

Form.     Since  their  time  difference  is  4  hr.  56  min. 
.2  sec.,  and  since  New  York  is  west,  its  time  is  earlier 
by  4  hr.  56  min.  .2  sec.,  or  3  min.  59.8  sec.  past  7  a.  m. 
Exercises. 

1.  Two  places  differ  in  time  6  hr.   7  min.  10  sec. ; 
required  their  difference  in  longitude. 

2.  Two  places  are  distant  from  each  other  1  quad- 
rant ;  what  is  their  time  difference  ?  When  it  is  10  a.  m. 
at  the  place  the   farther   west   what  time  is  it   at  the 
other?  When  it  is  1   o'clock   p.    m.    at   the  place  the 
farther  east  what  time  is  it  at  the  other? 

3.  A  man  travels  westward  5°17'11"  ;  is  his  watch 
too  fast  or  too  slow  and  how  much  ? 


98 
• 

4.  A  man  travels  until  his  watch  is  19  min.  55  sec. 
too  fast ;  has  he  traveled  east  or  west,  and  how  far  ? 

5.  A  man  travels  until  his  watch  is  too  slow  by  11 
hr.  5  min.  16  sec. ;  has   he   traveled   east  or  we^t,  and 
how  far? 

6.  The   longitude   of  St.   Louis  is   90°10',  while 
that  of  Cincinnati  is  84°26' ;  required  their  difference 
in  time.     When  it  is  noon  at  either  place  what  is  the 
time  at  the  other  ? 

7.  The  longitude  oi   Bangor  Me.,  is  68°45',  and 
that  of  San  Francisco  is   122°25' ;  required   their  time 
difference.     When  it  is  7  a.   m.   at   either   place  what 
time  is  it  at  the  other  ? 

8.  A  vessel   sailed  due  north   at  the  rate  of  14 
knots   per   hour,    while    the    sun    apparently   moved 
through  1  sextant    5°18' ;   how  long  did   she  sail,  and 
how  many  statute  miles  ? 

9.  f  Washington   is  in  longitude  77°2'48"  west  and 
Cincinnati  is  in  longitude  84°26'  west ;  when  it  is  6  a. 
m.  at  either  place  what  time  is  it  at  the  other? 

10.  If  my  watch  keeps  Terre  Haute  time  and  indi- 
cates 17  min.  10  sec.  past   1    o'clock   p.  m.   when   it  is 
noon  by  local  time  how  far   and  in  which  direction  am 
I  from  Terre  Haute  ? 

11.  A  man  travels  from  Pittsburg  in  longitude  80° 
2'  west,  until  his  watch  is  1   hour   and   45  minutes  too 
fast ;    how   far   and  in  which   direction  has   he  trav- 
eled ? 

12.  The   longitude   of    Galveston   is   94°50'   west 
while  Constantinople  is  in  longitude  28°49'  east  ;  when 
it  is  noon  at  either  place  what  time  is  it  at  the  other  ? 


13.  A  ship's  chronometer,  set   at   Greenwich,  indi- 
cates 5  hr.  45  min.  24  sec.  p.  m.  when  the  sun  is  on  the 
meridian;  required  the  longitude  of  the  vessel. 

14.  A  degree  of  longitude  in  the  latitude  of  Boston 
is  44sr  geographic  miles-.  How  many  more  statute  miles 
in  7°  of  longitude  at  the  equator  than  in  the  latitude  of 
Boston  ? 

15.  At  3  o'clock  and  35  min.  a.  m.  in  London,  what 
is  the  time  at  Boston  ? 

16.  At  5  o'clock  p.  m.  in  Eome,  what  time  is  it  in 
Terre  Haute  ? 

17.  At  noon  in  Pekin  what  time  is  it  in  San  Fran- 
cisco ? 

18.  How  many  degrees  of  east  or  west  longitude 
may  a  place  have  ?     Why  ? 

19.  What  is  the  difference  between  the  local  and 
''standard"  time  of  Terre  Haute  ? 

20.  When  it  is  10  o'clock  at  Indianapolis,  what  is 
"standard"  time  at  the  same  place  ? 

21.  When  it  is  4  p.  m.,  local  time,  at  Omaha,  what 
is  '  standard"  time  at  New  York  City  ? 

22.  When  it  is  5  hr.  15  min.  10  sec.  a.  m.,  "stand- 
ard" time  at  San  Francisco,  what  is  local  time  at  Pitts- 
burg  ? 

23.  At  6  hr,  30  min.,  a.  m.  local  time,  Jacksonville, 
Fla.,  what  is  the  "standard"  time  at  San  Antonio,  Texas  ? 

24.  When  it  is  noon  at  Cincinnati,  local  time,  is  it 
before  or  afternoon  and  how  much  by  "standard"  time 
at  Cleveland,    Toledo,    Detroit,   Ft.     Wayne,   Mobile, 
Eichmond  ? 


100 

SECTION  X. 
ABBAS  AND  yOLUMES. 
174.     Areas. 

Example  1.     A  room  is  12  ft.  long  and  8  ft.  wide  ; 
how  many  sq.  ft.  are  in  the  floor  ? 

Solution. 

A  surface  1  ft.  1.  and  1  ft.  w.  =  1  sq.  ft. 
A  surface  12  ft.  1.  and  1  ft.  w.  =  12  times  1  sq. 
ft.  =  12  sq.  ft. 

A*  surface  12  ft.  1.  and  8  ft.  w.  =  8  times  12  sq. 
ft.  =  96  sq.  ft. 
.-.  etc. 

Remark.  Length,  width  and  area  are  always  expressed  in 
concrete  units,  hence  length  cannot  be  multiplied  by  width  ; 
even  if  such^multiplication  were  possible,  the  product  would 
not  be  area,  which  is  different  in  name  from  the  multiplicand. 

If,  however,  the  numbers  representing  the  dimensions  of  a 
parallelogram  be  multiplied  together,  the  product  is  the  number 
representing  the  square  units  in  the  given  surface. 

Example  2.  A  lot  contains  192  sq.  rd.  and  is  16  rd. 
long ;  how  wide  is  it  ? 

Solution.      A  surface  1  rd.  1.  and  1  rd.  wide  =  1  sq.  rd. 
A  surface  16  rd.  1.  and  1  rd.  wide  =  16  times 

1  sq.  rd.  =  16  sq.  rd. 
A  surface  16  rd.  1.  must  be  as  many  rd.  wide 

to  contain  192  sq.  rd.  as  192   sq.   rd.  are 

times  16  sq.  rd.  which  =  12. 

.-.  The  lot  is  12  rd.  wide. 

Remark.  If  the  number  of  square  units  in  a  parallelogram  be 
divided  by  the  number  of  corresponding  linear  units  in  either 
dimension,  the  quotient  is  the  number  of  linear  units  in  the 
other  dimension. 

Example  3.     The  base  of  a  plane  triangle   is  six 
inches  and  its  altitude  is  4  inches  ;  what  is  its  area  ? 


101 

(1.)  The  area  of  a  plane  triangle  is  one-half  the 
area  of  a  parallelogram  having  the  same  base  and  alti- 
tude as  the  triangle;  hence  find  the  area  of  such  paral- 
lelogram as  in  Ex.  1,  and  divide  it  by  2. 

(2.)  If  the  number  of  linear  units  in  the  base  or 
altitude  of  a  plane  triangle  be  multiplied  by  one-half 
the  number  of  like  linear  units  in  the  other  dimension, 
the  product  is  the  number  of  corresponding  square 
units  in  the  triangle. 

Example  4.  The  diameter  of  a  circle  is  14  inches; 
what  is  its  area? 

(1.)  The  circumference  of  a  circle  is  nearly  3-iJ- 
times  the  diameter. 

(2.)  Any  circle  is  practically  equal  to  a  rectangle 
whose  length  is  the  semi-circumference  and  whose 
width  is  the  radius  of  the  given  circle. 

Solution.  3|  times  14  inches  =  44  in.  —  the  ap- 
proximate circumference  of  the  given  circle. 

Its  equivalent  rectangle  is,  therefore  22  in.  by  7 
in.  7  times  22  =  154.  .'.  the  approximate  area  of  the 
given  circle  is  154  sq.  in. 

(3.)  The  area  of  any  circle  is  .7854  of  the  area  of 
its  circumscribed  square. 

Each  side  ot  the  square  that  circumscribes  the 
given  circle  is  14  inches,  the  diameter  of  the  circle. 
The  area  of  the  square  is  found  by  multiplying  14  by 
14  and  giving  the  denomination  square  inches  to  the 
product.  .7854  of  this  product  equals  154-f-  sq.  in., 
the  area  of  the  given  circle. 

The  rule  is  usually  formulated  thus: — To  find  the 
area  of  a  circle:  Multiply  the  square  of  the  diameter  by 
.7854. 


102 

175.     Volumes. 

Example  1.     A  common   brick  is  8  in.  long,  4  in. 
wide  and  2  in.  thick ;  how  many  cu.  in.  does  it  contain? 

Solution. 

A  solid  1  in.  1.  1  in.  w.  1  in.  th.  =  1  cu.  in. 
A  solid  8  in.  1.  1  in.  w.  1  in.  th.  =  8  times  1  cu.  in.  =8 

cu.  in. 
A  solid  8  in.  1. 4  in.  w.  1  in.  th.  =  4  times  8  cu.  in.  =  32 

cu.  in. 
A  solid  8  in.  1.  4  in.  w.  2  in.  th.  =  2  times  32  cu.  in.— 

64  cu.  in. 
/.  a  brick  contains  64  cu.  in. 

Remark.  If  the  numbers*  representing  the  three  dimensions 
of  a  rectangular  solid  be  multiplied  together,  the  product  is 
the  number  representing  the  corresponding  volume  units  in  the 
given  solid, 

Example  2.     A  bin  containing  315  cu.  ft.  is  9  ft. 
long  and  5  ft,  wide,  how  deep  is  it  ? 

Solution.    - 

A  solid  1  ft.  1.  1  ft,  w.  1  ft.  d.  =  1  cu.  ft. 
A  solid  9  ft.  1.  1  ft.  w.  1  ft.  d.  =  9  times  1  cu.  ft.  =  9 

cu.  ft. 
A  solid  9  ft.  1.  5  ft.  w.  1  ft.  d.  =  5  times  9  cu  ft.  =  45 

cu.  ft. 

A  solid  9  ft.  1.  5  ft.  w.  must  be  as  many  ft.  deep  to  con- 
tain 315  cu.  ft.  as  315  cu.  ft.  are  times  45  cu.  ft.  which 
equal  7.  .'.  the  bin  is  7  ft.  deep* 

Remark.  If  the  number  representing  the  volume  of  a  rectan- 
gular solid  be  divided  by  the  product  of  the  numbers  represent- 
ing two  of  its  dimensions,  the  quotient  is  the  number  repre- 
senting its  third  dimension. 

Exercises. 

1.  How   many  square  feet  in  a  surface  12  ft.  by 
5ft.? 

2.  How  many  sq.  yd.  in  a  lot  15  yd.  by  11  yd.? 


103 

3.  A  field  is  40  rods  long  and    18  rods  wide;  how 
many  acres  in  it  ? 

4.  A  room  is  15  ft.  6  in.  long  by  24  ft.  4  in,  wide  ; 
how  many  square  yards  in  the  ceiling  ? 

5.  A  floor  contains  192  sq.   yd,    and   is  12  yards 
wide  ;  how  long  is  it  ? 

6.  A  field  contains   18  acres  and  is   60  rods  long  • 
how  wide  is  it? 

7.  A  brick  walk  is  f  yd.  wide  ;  how  long  must  it 
be  to  contain  25  square  yards  ? 

8.  A  triangular  wall  is  6  ft.   long  and  5£  ft.  high 
at  its  widest  end  ;  what  is  the  area  of  one  side  ? 

9.  A  block  is  5  in.  by  3  in.  by  2   in. ;   what   is  its 
volume? 

10.  A  piece  of  stone  is  7  ft.  long,  5-J-.  ft.  wide  by  3£ 
ft.  thick  ;  how  many  cu.  ft.  in  it? 

11.  A  bin  is  8  ft.  by  6   ft.    by   4J  ft.  ;   how   many 
bushels  of  wheat  will  it  hold? 

12.  A  cistern  is  9  ft.  by  5  ft.  by  5  ft. ;  how  many 
gallons  does  it  hold  ? 

13.  How  many  perch  of  masonry  in  a  wall  350  ft 
by  18  ft.  by  2  ft.  ? 

14.  A  block  contains  64  cu.  ft.,    and   is  4  ft.  wide 
and  -2  ft.  thick  ;  how  long  is  it  ? 

15.  A  box  is  5  ft.  long  and   3   ft.    wide  ;   how  deep 
must  it  be  to  contain  60  cu.  ft.  ? 

16.  A  tank  is  10  ft. -Long  and  4  ft.   wide';  how  deep 
must  it  be  to  hold  3  tons  of  water  ? 

17.  A  haystack  is  36  ft.  long  and  9  ft.  wide  ;  what 
is  its  hight  if  it  contain  11  tons  of  hay  ? 

18".  A  horse  is  tied  to   a   stake    by    a   rope    100ft. 
long  •  over  how  much  ground  can  he  walk  ? 


104 

SECTION  XI. 

THE  DECIMAL  SYSTEM  OF  MEASURES. 

Remark.  The  history  of  the  decimal  system  of  measures  and 
numerous  facts  and  arguments  showing  its  simplicity,  are  found 
in  the  publications  of  the  American  Metric  Bureau  of  Boston. 

The  purpose  of  this  section  is  to  exhibit  the  system  and  to 
present  methods  of  computing  by  means  of  it. 

176.  Primary  Units. 

Of  length,    — the    meter.         marked  m. 

'*   Capacity, —  "    liter  (leeter.)     "  1. 

"   weight,    —   "    gram,  "  g.  • 

Remarks.  I.  The  ar  is  the  unit  for  measuring  land.  Other 
surfaces  are  measured  by  the  sq.  m.  or  by  decimal  parts  or  deci- 
mal multiples  thereof.  The  ar  is  marked  a. 

2.  The  ster  is  the  unit  for  measuring  wood.     Other  volumes 
are  measured  by  the  cubic  meter  or  by  decimal  parts  or  deci- 
mal multiples  thereof.     The  ster  is  marked  s. 

3.  In  measuring  great  weights  the  quintal  and  the  tonneau 
(Metric  ton)  are  used.    These  are  marked  Q.   and  T.,  respect- 
ively. 

177.  Secondary  Units. 

Each  of  the  secondary  units  used  is  either  a  deci- 
mal part  or  a  decimal  multiple  of  a  primary  unit,  and  is 
designated  by  using  one  of  the  following  prefixes  with 
the  name  of  a  primary  unit. 

Decimal  parts.  Decimal  Multiples. 

Dgei,    meaning      .1.  Deka,    meaning         10. 

Cent*,  .01.  Hgkto,         "  100. 

Milli,  .001.  Kflo,  1000. 

Myria,  10000. 

Remarks.  1.  In  combining  any  one  of  the  foregoing  prefixes 
with  the  name  of  a  primary  unit,  each  word  retains  its  own 
pronunciation  and  accent. 

2.  In  abbreviating  a  word  formed  by  combining  a  prefix  with 
the  name  of  a  primary  unit,  it  is  customary  to  use  the  initial 
letter  of  each  word,  using  a  small  letter  to  designate  the  .prima- 
ry unit  or  any  of  its  decimal  parts,  and  a  capital  letter  to  desig- 
nate any  of  its  decimal  multiples. 


105 

Scales. 
178.     Of  Length  Measure. 

10  10  10  10  10  JO  10 

Mm.     Km.        Hm.      Dm.        m.         dm.        cm.        mm. 
11  111111 


179.     Of  Surface  Measure. 

100     100      100      100     100      100     100 

Sq.  Mm.  sq.  Km.  sq.  Hm.  sq.  Dm.  sq.  m.  sq.  din.  sq.  cm.  sq.  mm. 
11111111 


180.     Of  Land  Measure. 

10  10  10  10  10  10  10 

Ma.     Ka.        Ha.        Da.          a.          da.         ca.         ma. 
11111111 


181.    Of  Volume  Measure, 

1000     1000     1000     1000    1000    1000     1000 

Cu.  Mm.  cu.  Km.  cu.  Hm.  cu.  Dm.  cu.  m.  cu.  dm.  cu.  cm.  cu.  mm. 
1  1  1  11111 


182.     Of  Wood  Measure. 

Remark.     Only  three  units  are  used. 

10  10 

Ds.       s.        ds. 
1         1         1 


183.     Of  Liquid  and  Dry  Measure. 

10  10  10  10  10  10  10 

Ml.          Kl.      HI.      Dl.        1.        dl.        cl.      ml. 
1  1111111 


184.     Of  Weight. 

10           10           10  10           10           10           10           10           10 

T.      Q.      Mg.      Kg.  Hg.     Dg.       g.       dg.      eg.      mg 

lilt  11111 


106 

Relation  of  Decimal  and  Common  Measures. 
185.    Of  Length.  186.  Of  Surface. 

1  cm.   —  .3937  in.          1  sq.  dm.  =  15£  sq.  in. 

1  dm.  =    *    3.937  in.  1  sq.  m.    =  1550  sq.  in. 

1m.      =      39.37  in.  _  ( 119.6  sq.  yd. 

1  Dm.  =      32.8  ft.  -  {  1  sq.  Dm. 

H     m=.  328  ft.  1  in:  1  Ha.  =  2.47  acres. 

1  Km.  =  3280  ft.  —  f  mi.,  nearly. 
1  Mm.  6.2137  miles. 


187.    Of  Volume. 

1  cu.  cm 

=        .06  cu.  in. 

1  eu.  dm. 

=     61  cu.  in. 

1  cu.  m. 

\       C  35.317  cu.  ft., 

or 

j  '  —  i            ^^* 

1  ster. 

)       (    1.308  cu.  yd. 

188.    Of  Capacity. 

1  ml.  =    1  cu.  cm.     = 

.27  fl.  dr.  =  .061  cu.  in. 

1  cl.   =  10  cu.  cm.     — 

.338  fl.  oz,  =.61     "     " 

1  dl.   =  .1  cu.  dm.     = 

.845  gi.  =6.1         "     " 

11.     .=    1  cu.  dm.     = 

1.05  qt.  liq.  =  .9  qt.  dry. 

1  Dl.  =  10  cu.  dm.     =. 

2.64  gal.  liq.  =  9'.08  qt.  dry. 

1  HI.  =  .1  cu.  m.       = 

26.41  gal.  liq.  =  2.76  bu. 

1  Kl.  =  1  cu.  m.       = 

264.17   gal.  liq.  =  1.308  cu.  yd. 

189.    Of  Weight. 

1  mg.=rthe  wt.  of  1  cu 

.  mm.  of  water=        .015  gr.Av. 

1  eg.  =  «     "     "  10  " 

"      "      "    =        .154  "     " 

1  dg.  ==  «    «    "  '.1  «' 

cm.    »«       "     =      1.54     '     " 

1  g.    =  "    4<    "     1  4/ 

"      "      "     =     15.43     "     " 

1  Dg.—   "     "     "  10  " 

«      "      ««     =        .35   oz.   " 

1  Hg.=  "     "     "     1  dl 

,     «      «       "    =      3.53     "     " 

1  Kg.=    «      «      "      1    1. 

•c      "     —      2.2    Ib.    " 

1  Mg.=   "     "     "  10  " 

"      "     —     22.04     u     " 

1  Q.  =    "     "     "     1  HI.         "      "    =  220.4      «     " 

IT.   =   "     "     ll     1  ci 

n.  m.             "     =2204.b 

107 

Remark.  The  fractions  in  the  above  tables  of  relative  value 
are  not  exact,  but  they  are  sufficiently  approximate  to  meet 
most  applications.  Indeed  where  no  great  accuracy  is  required 
it  is  sufficient  to  use  the  following  table  of — 

190.  Approximate  Values. 

1  dm.  =  about  4  in. 

1m.  =  "    39i  in. 

5m.  =  "      1    rod. 

1  Km.  "        f  mi. 

1  sq.  m.  "    lOf  sq.  ft. 

1  Ha.  =  "      2|  acres. 

1  cu.  m.  or  ster.  =  u      1J-  cu.  yd.,  or  i  cord. 

11.  =  u      1    qt. 

1  HI.  =  "      2|bu. 

1  g.  =  «    15J  gr. 

1  Kg.  ==  «      2*  lb. 

IT.  •  =  a  2200  lb. 

191.  Table  of  Specific  Gravities;  Water  Being  I. 

Air,                            .001  Ice,  .93 

Alcohol,  pure,           .79  Iron,  cast,  7.1 

"     commercial,  .83                  "     wrought,  7.7 

Brass,                       7.6  Lead,.  11.35 

Brine,                       1.04  Lime,  1.8 

Coal,  soft,                  1.25  Marble,  2.7 

"     hard,                1.5  Mercury,  13.6. 

Copper,                     9.  Milk,  1.03 

Cork,                         i-  Silver,  10.5 

Gold,  19.2  Zinc,  7.1 

192.  Important  Facts. 

1.  The  ar  is  an  area  equivalent  to  1  sq.  Dm. 

2.  The  ster  (pronounced  stair}  is  a  volume  equiv- 
alent to  1  cu.  m. 

3.  The  liter  is  a  volume  equivalent  to  1  cu.  dm. 


108 

4.  The  gram  is«the  weight  of  a  cu.  cm.  of  distilled 
water  at  its  greatest  density,  i.  e.,  at  4°  centigrade,  or 
39.2°  Fahrenheit,   The  water  is  balanced  in  a  vacuum. 

5.  1  liter,  or  1  cu.  dm.  of  distilled  water  weighs  1 
kilogram,  while  1  cu.  m.  of  water  weighs  1  metric  ton. 

6.  The  legal  letter  weight  in  the  United  States  is 
i  oz.  Av.     The  Postmaster   General  is   authorized  to 
substitute  a  15  gram  weight  for  the  £  oz.  weight  in  all 
postoffices  that  exchange  mails  with  foreign  nations 
and  in  other  postoffices  at  his  discretion. 

7.  A  freshly  coined  nickel  5  cent  piece  (not  the 
new  nickel),  is  2  cm.  in  diameter  and  weighs  5  grams. 
The   silver  coinage  of  the  United  States  is  worth   4 
cents  per  gram.     Three  5  cent  nickels  or  60  cents  in 
silver  coin  constitute  one  letter  weight. 

193.     Applications  of  Decimal  Measures. 

Length. 

Write  and  read    1     m.  as  dm.  cm.  mm.  Dm.  Hm.  Km. 
(t        n        it   254m         t£      "      "        u        u     li 

<c  It  it  9  (j4jY)  a  tt  it  It  it  it 

tt  It  tt  Q    f\Yf\  l  "  "  "  "  " 

It  it  it              K     U  it  U  It  tt  U  tt 

U  tt  It  ft     TVIVQ  "  '*  u  U  il  " 

u        "  15  Dm.  "m  "      "       "  *'      " 

34m.-f42dm.H-134  cm.  =  how  many  m.?  dm.?  cm.?  etc 

Capacity. 
Write  and  read     1     1.  as  dl.  cl.  ml.  Dl.  HI.  Kl.  Ml. 

a    •    34  5  1  "      "      "      "        "        ll       li 


^          t:       ((       U       ((          ti         a         li 

it       ti       OK  Q]  14     «     it     «        it       ii       it 


2 


109 


Weight. 
Write  and  read     1     g.  as  dg.  eg.  mg.  Dg.  Hg.  Kg.  Q.T. 


" 


3.4  Cg 

u  a         u         P      o        .       a      u        a        u        u        u 


69  Kg. 


"      " 


[Exercises.] 

Remarks.  1.  Areas  and  volumes  are  found  as  in  the  common 
measures. 

2.  In  finding  the  capacity  of  a  given  volume  it  is  necessary 
to  remember  that  a  cu.  dm.  ==  1  liter. 

3.  In  finding  the  weight  of  a  given  volume  or  capacity  of  a 
substance,  it  is  necessary  to  remember  that  a  cu.  cm.  of  dis- 
tilled water  weighs  1  g.,  that  1  cu.  dm.  of  distilled  water  weighs 
1  Kg  and  fills  a  liter  cup,  and  that  1  cu.  m.  of  distilled  water 
weighs   one  metric  ton.     If  the  weight  of  a  substance  other 
than  water  be  required,  we  find  the  weight  of  an  equal  vol- 
ume of  water  and  multiply  it  by  the  specific  gravity  of  the 
substance. 

Example  1.  A  tank  is  4  m.  long,  3  m.  wide  and  2.5 
m.  deep;  how  many  Kg.  of  brine  will  fill  it? 

Solution.    4x3x2."5  =  30  —  the  number  of  cu.  m. 
in  the  tank. 

Since  1  cu.  m.  of  distilled  water  weighs  1  metric  ton, 
30  cu.  m.  of  distilled  water  weigh  30  metric  tons;  and 
since  1  T.  =  1000  Kg.,  30  T.  =  30000  Kg.  =  the  weight 
of  the  water  necessary  to  fill  the  tank. 

Since  brine  is  1.04  times  as  heavy  as  water,  the 
weight  of  this  volume  of  brine  =  1.04  times  30000  Kg., 
or  31200  Kg.  /.  etc. 

2.  Find   the   area  of  tbe  walls  of  a  room  6.2  m. 
long,  5.05  m.  wide  and  3.5  m.  high. 

3.  How  many  rolls  of  paper  45  cm.  wide  and  8  rn. 
long,  allowing  11.2  sq.  m.  for  openings  will  be  required 
to  paper  those  walls  ? 


110 

4.  Find  the  cost  of  plastering  the  room  at  50/  per 
sq.  m. 

5.  How  many  sq.  m.  in  a  board  8  m.  by  25  m.?  5 
m.  by  25  cm.?  7  dm.  by  156  mm.? 

6.  If  wood  is  cut  into  120  cm.  lengths,  and  a  pile 
is  43.7  m.  long  and  1.6  m.  high,  how  many  sters  does  it 
contain  ? 

7.  A  bin   is  11.4  m.  long,  4.15  m.  wide  and  2.8  m. 
deep ;  how  many  hektoliters  of  barley  does  it  hold  ? 

8.  If  the   specific  gravity  of  grain  is  .81,  what  is 
the  weight  of  the  grain  that  fills  the  bin  ? 

9.  A  vat  is  186  cm.  long,  7.7  m.  wide  and  48  dm. 
deep ;  how  many  tonneaus  of  water  does  it  hold  ? 

10.  What  is  the  weight  in  kilograms,  of  a  en.  cm. 
of  water?     Of  a  HI.  of  water?  Of  Mercury?  Of  milk? 
Of  lime  ?    Of  a  cu.  m.  of  cork?    Of  brass?    Of  zinc? 

11.  What  weight  of  water  will  fill  a  vat  92  cm.  by 
76  cm.  by  4.2  dm.?     What  weight  of   milk  will  fill  it? 

12.  If  the  above  vat  be  filled  with  brine  weighing 
1.04  Kg.  per  liter,  required  the  weight  of  the  brine? 

13.  How  many   liters  of  air  in  a   room  6.3  m.  by 
5.17  m.  by  2.9  m.  ?     What  is   the   weight   of  the  air  in 
grams  ?     In  kilograms  ? 

14.  How  many  pills  of  .36  g.  each,  can  be   made 
from  a  mass  weighing  .72  Kg.  ? 

15.  What  is  the  weight  of  7.1  HI.  of  pure  alcohol? 

16.  An  irregularly  shaped  mass  of  copper  displaces 
.88  1.  of  water;  what  is  its  weight  in  kilograms? 

17.  A  piece  of  iron  125  cni.  long,  56  cm.  wide,  and 
6.2  cm.  thick  weighs  256.4  kilograms;    what  is  the  spe- 
cific gravity  of  the  iron  ? 


Ill 

18.  A  piece  of  ore  weighing  5.6  kilograms,  weighs 
in  water  only  3.12  kilograms ;  what  is  its  s.  g.  ? 

19.  If  a  tap  running  2.7  1.  per  min.,   fill  a  tub  in  14 
minutes,  how  long  would  it  take   a   tap  running  4.2  1. 
per  min.  to  fill  it? 

20.  A  cistern  will  hold  18  tonneaus   of  rain  water; 
what  depth  of  rain  must  fall  upon  a  flat  roof  20  m.  long 
by  15m.  wide,  to  fill  the  cistern  ?  • 

21.  Find  the  weight  in  Kg.  of  a  block  of  ice  4.5  m. 
by  3.2  m.  by  2  dm. 

22.  What  is  the  weight  of  a  bar  of  lead  2.3  dm.  by 
2  cm.  by  1.2  cm.  ? 

23.  What  is  the  weight  of  a  piece  of  copper  4  dm. 
by  1.6  dm.  by  3  cm.  ? 

24.  What  is  the  value  of  a  piece   of  silver  3.2  dm. 
by  7  cm.  by  2  cm.  at  3^  per  gram  ? 

25.  How  many   bullets,    each    weighing   2.7  deka- 
grams can  be  made  from  a  cubical   block  of  lead  whose 
edge  is  .74  dm.  ? 

26.  What  is  the  s.  g.  of  a  substance  that  weighs  20 
Kg.  in  air  and  18  Kg.  in  water  ? 

27.  How  many  jets,  each  burning  110  liters  of  gas 
per  hour  for  21  hours  each  night,   would    consume  the 
contents  of  a   gasometer   containing   300000   cu.  m.  of 
gas? 

28.  What  is  the  weight  of  a  block  of  marble  .72m. 
by  12  dm.  by  1.2  dm.  ? 

29.  A  box  is  2.3  m.  by  9  dm.  by  25  cm.  ;  how  many 
liters  does  it  hold  ? 

30.  If  a  wheel  is   90   cm.    in   circumference,    how 
many  times  does  it  revolve  in  going  5  kilometers  ? 


112 

31.  If  a  cu.  cm,  of  ore  weigh    6.2  g.,  required    its 

8.    g. 

32.  How  many  meters  of  carpeting  .7  m.  wide,  will 
cover  a  floor  6m.    by  7  m.,   the   strips   extending  the 
longer  way  of  the  floor  ? 

33.  Required  the  area  of  a   circle   3.2  in.  in  diam- 
eter. 

34.  If  a  cu.  m.  of  earth    weigh    1268  Kg.,  what  is 
its  s.  g.? 

35.  A  cistern  is  4  m.  deep  by  3  m.    wide,  how  long 
must  it  be  to  hold  60  metric  tons  of  water  ? 

36.  What  is  the  Troy   weight  of  3  cu.  dm.  of  mer- 
cury ? 


SECTION  XII. 
PERCENTAGE. 

194.  Percentage  is  a  method  of  computing  by  hun- 
dredths. 

The  Terms  Employed. 

195.  The  Base.     The  base  is  the  number  of  which  a 
number  of  hundredths  is  reckoned. 

196.  The  Rate  Pep  Cent.     The  rate  per  cent,  is  the 
fraction  which  indicates  the  number  of  hundredths  in- 
volved. 

197.  The  Percentage.      The  percentage  is  a  number 
that  bears  the  same  relation  to   the   base  that  the  rate 
%   does  to  1.     It  is   a   number   of  times   .01    of   the 
base  if  the  rate  per  %  is  .01  or  more;  it  is  a  part  of  .01 
of  the  base  if  the  rate%  is  less  than  .01. 


113 

Remarks.  1.  The  term  amount  is  applied  to  the  sum  of  the 
base  and  ,the  percentage.  . 

The  base  is  TTTO-  of  itself  and  the  percentage  is  a  number  of 
hundredths  of  the  base,  hence  their  sum,  the  amount,  is  a 
number  of  hundredths  of  the  base,  and  comes  within  the  defi- 
nition of  the  percentage. 

The  term  difference  is  applied  to  the  part  of  the  base  that  re- 
mains after  the  withdrawal  of  the  percentage  from  the  base. 
The  difference  is,  therefore,  a  number  of  hundredths  of  the 
base  and  comes  within  the  definition  of  the  percentage. 

198.  Relations  of  Percentage.  (1.)  The  percentage, 
being  a  number  of  hundredlhs  of  the  base,  is  found  by 
obtaining  the  number  of  hundredths  of  the  base  that 
the  rate  per  cent,  indicates  ;  i.  e.  by  multiplying  the 
base  by  the  rate  per  cent.  The  base  and  the  rate  per 
cent,  are  thus  seen  to  be  1  tic  tors  of  the  percentage. 

Percentage  is  related  to  multiplication  in  the  sig- 
nification of  its  terms ;  the  base  being  multiplicand,  the 
rate  per  cent,  being  multiplier  and  the  percentage  being 
product. 

(2.)  Principle.  If  two  or  more  factors  are  given, 
their  product  is  found  by  multiplying  the  factors  to- 
gether. 

Principle.  If  the  product  of  two  factors  and  one 
of  them  be  given,  the  other  is  found  by  dividing  the 
product  by  the  given  factor. 

In  the  light  of  one  or  the  other  of  the  above  prin- 
ciples of  factoring  is  seen  the  process  to  be  performed 
in  finding  any  one  of  the  three  terms  of  percentage  it 
the  other  two  are  known. 

Percentage  is  related  to  factoring  in  the  principles 
which  determine  the  processes  to  be  performed. 

(3.)  The  rate  per  cent,  is  always  given  in  the  de- 
nomination of  hundredths. 

Percentage  is  related  to  fractions  in  that  one  of  its 
terms  (the  rate  per  cent.)  is  a  fraction. 


114 

General  Cases  of  Percentage. 

Remark.    All  special  problems  in    percentage  may  be  classi- 
fied under  three  cases. 

199. 

CASE  I. 

Given  the  base  and  the  rate  per  cent,  to  find  the 
percentage. 

Solution.  Since  the  percentage  is  the  product  of 
the  base  and  the  rate  per  cent.,  the  percentage  is  found 
by  multiplying  the  base  by  the  rate  per  cent. 

200. 

CASE  II. 

Given  the  base  and  the  percentage  to  find  the  rate 
per  cent. 

Solution.  Since  the  percentage  is  the  product  of 
the  base  and  the  rate  per  cent.,  the  rate  per  cent,  is 
found  by  dividing  the  percentage  by  the  base,  expres- 
sing the  quotient  in  the  denomination  of  hundredths. 

201. 

CASE  III. 

Given  the  percentage  and  the  rate  per  cent,  to  find 
the  base. 

Solution.  Since  the  percentage  is  the  product  of 
the  base  and  the  rate  per  cent,,  the  base  is  found  by 
dividing  the  percentage  by  the  rate  per  cent. 

Remark.  It  is  observed  that  Case  I  is  solved  by  multiplica- 
tion in  the  light  of  the  first  of  the  two  principles  stated  under 
the  second  relation  of  percentage,  while  Cases  II  and  III  are 
both  solved  by  division  in  accordance  with  the  second  of  the 
two  principles  referred  to.  On  the  basis  of  processes  em 
ployed,  two  cases  will  be  found  to  embrace  all  problems  in  per- 
centage. 


115 

202.     Forms  of  Solution.     (1.)  What  is  1%  of  35  ? 

First  form.  We  have  given  the  base,  35,  and  the 
rate  per  cent.,  .07,  to  find  the  percentage.  Since  the 
percentage  is  the  product  ol  the  base  and  the  rate  per 
cent.,  the  percentage  in  this  example,  is  found  by  mul- 
tiplying 35  by  .07.  The  product,  2.45,  is  the  required 
percentage. 

Second  form.     \%  of  35  =  .01  of  35  =  .35. 

7%  of  35  =  7  times  .35  =  2.45. 
.-.7%  of  35  =  2.45. 

(2.)  45  =  what  %  of  75? 

First  form.  We  have  given  the  base,  75,  and  the 
percentage,  4.5,  to  find  the  rate  per  cent.  Since  the 
percentage  is  the  product  of  the  base  and  the  rate  per 
cent.,  the  rate  per  cent.,  in  this  example,  is  found  by 
dividing  4.5  by  75,  expressing  the  quotient  as  hun- 
dredths.  The  quotient,  .06,  is  the  required  rate  per 
cent. 

Second  form.  \%  of  75  =  .01  of  75  =  .75  ;  hence 
4.5  equal  as  many  times  \%  of  75  as  .75  is  contained 
times  in  4.5  which  =  6.  6  times  1%  =  6%.  Therefore 
4.5  =6%  of  75. 

Third  form,  v  1  =  -fe  ot  75,  4.5  =  4.5  times  ^  of 
75  =  ^f  of  75.  Multiplying  both  terms  of  ^f  by  1 J, 
we  have  .06.  /.  4.5  =  Q%  of  75. 

(3.)     What  per'cent.  of  a  number  is  ^  of  it  ? 

Solution.  -£$  =  -f/-$.  Hence  -fa  of  a  number  = 
.35,  or  35%  of  it. 

(4.)     36  equal  9%  of  what  number? 

First  form.  We  have  given  the  percentage,  36,  and 
the  rate  per  cent.,  .09,  to  find  the  base.  Since  the  per- 
centage is  the  product  of  the  base  and  the  rate  per 


116 

cent.,  the  base  in  this  example,  is  found  by  dividing  36 
by  .09.     The  quotient,  400,  is  the  required  base. 
(5.)     48  =  20%  more  than  what  number? 
First  form.     Since  48  =  20%  more  than  some  num- 
ber, 48  =  120%  of  that  number.      We  now  have  given 
the  percentage,  48,  and  the  rate  per  cent.,  1.20,  to  find 
the  base.     Since  the  percentage  is   the  product  ot  the 
base  and  the  rate  per  cent.,  the  base  in  this  example,  is 
found  by  dividing  48  by  1  20.     The  quotient,  40,  is  the 
required  number. 

Second  form.  Since  48  =  20%  more  than  some 
number,  48  ==  120%  of  that  number;  and  1%  of  the 
number  must  =  -^  of  48  =  ,4,  and  100%  of  the  num- 
ber must  ==  100  times  ,4  —40,  .',48  =  20%  more 
than  40, 

Third  form.  Since  20%  of  a  number  =  \  of  it, 
then  48  is  f  of  some  number, 

£  of  that  number  must  =  -J-  of  48  =  8;  and 
|  of  the  number  =  5  times  8  =  40. 
,:;  48  =  20%  more  than  40. 
(6.)     f  m  10%  less  than  what  number? 
First  form.     Since  f  =  10%  less  than   some  num- 
ber, f  —  90%  of  that  number, 

We  now  have  given  the  percentage,  f,  and  the  rate 
%,.90,  to  find  the  base. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent.,  the  base  in  this  example  is  found 
by  dividing  f  by  ,90.  The  quotient,  f ,  is  the  required 
base, 

Second  form.  Since  10%  less  than  some  number—  f 

90%  of  the  number  must  =  f  ; 
and    1%   «     «         «  "       =1foofi==TiiF« 

"  100%   "     "         "  " 

.-.  |=10%  less  thanf. 


117 

Exercises. 
J5ose.  Rate,  Percentage. 

1,  45,  5  %          ? 

2,  15.  9  «          ? 

3,  1.25  12 «          ? 

4,  45  *  «          ? 

5,  |  f  "          ? 

6,  60 

7,  450 

8,  1250 

9,  2,5 
10,  I 

n,  i 

12.  15 

13.  ? 

14.  ? 

15.  ? 

16.  ? 

17.  ? 

18.  ? 

19.  ? 

20.  ? 

21.  ? 

22.  ? 

23.  ? 

24.  ? 

25.  f 

26.  235  11   "         ? 

27.  4.5  2.5  "        ? 

28.  .004  .04  "         ? 

29.  54  .3   u         ? 

30.  160  ?  48 

31.  515  ?  3 

32.  4  ?  .005 


? 

15 

? 

45 

? 

600 

? 

5 

? 

I 

9 

.875 

p 

2,25 

4  % 

48 

15  " 

750 

24  " 

96 

1  " 

t 

1  " 

.375 

661  " 

540 

75  « 

16 

110" 

64 

125  u 

150 

134  % 

2680 

101£  " 

.  140 

118  " 

236 

a 


6 


118 

Applications  of  Percentage. 

203.  There  are  two   classes  of  applications  of  per- 
centage.    The  first  class  includes  all  those  problems  in 
which  the  percentage  is   the   product   of  the  base  and 
the  rate  per  cent. 

The  second  class  includes  those  problems  in  which 
the  percentage  is  the  product  of  three  factors,  viz.f  the 
base,  the  rate  per  cent.,  and  the  number  representing 
the  time  (in  years)  involved  in  the  transaction  under 
consideration. 

204.  The  principal   applications  of  the   first  class 
are — Profit   and   Loss,    Commission    and    Brokerage, 
Stocks,  Insurance,  Taxes  and  Customs. 

The  principal  applications  of  the  second  class  are — 
Interest,  Discount,  Bonds,  Exchange,  Equation  of  Pay- 
ments and  Accounts, 

Remarks.  1.  In  every  problem  of  the  first  class  the  three 
terms  of  percentage  are  represented. 

2.  Many  of  the  problems  to  be  solved  are  compound,  being 
composed  of  two  or  more  problems,  some  of  which  may  not 
involve  percentage. 


Applications  of  the  First  Class. 

205.  Profit  and  Loss.     (1.)  The  number  on  which 
the  gain  or  loss  is  estimated  is  the  base.     In   most  ex- 
amples the  cost  price  is  the  base. 

(2.)  The  gain,  loss  or  selling  price  is  the  percent- 
age. 

(3.)  The  rate  of  gain,  rate  of  loss  or  rate  of  selling 
is  rate  per  cent. 

206.  Forms  of  Solution. 

Example  1.  A  man  paid  $110  for  a  horse  and 
sold  it  at  a  profit  of  20  per  cent. ;  required  the  gain. 


IS 


110 

First  form.  We  have  given  the  cost,  $110,  which 
base,  and  the  rate  of  gain,  .20,  which  is  rate  per  cent., 
to  find  the  gain,  which  is  percentage. 

Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent.,  the  percentage  in  this  example 
is  found  by  multiplying  $110  by  .20.  The  product, 
$22,  is  the  required  percentage,  which  is  the  gain. 

Second  form.     The  gain  is  estimated  on  the  cost. 

1%  of  $110  =  .01  of  $110  =$1.10,  and 
20%  of  $110  =  20  times  $1.10  =$22. 
.-.  The  gain  was  $22. 

Third  form,     v  20%  of  a  number  =  |  of  it, 

20%  of  $110  =  i  of  110  =  $22. 
/.  the  gain  was  $22. 

Example  2.  If  a  merchant  pay  15  cents  per  yard 
for  muslin,  for  how  much  does  he  sell  it  to  lose  25%  ? 

First  form.  Since  he  sells  at  25%  below  cost,  he 
sells  for  75%  of  the  cost  ;  and  we  have  given  the  cost 
price,  15/,which  is  base,  and  the  rate  of  of  selling,  .75, 
which  is  rate  %,  to  find  the  selling  price,  which  is  theper- 
centage.  Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent  ,  the  percentage  in  'this  example 
is  found  by  multiplying  15/  by  .75;  the  product,  \\\'fi 
is  the  required  percentage,  which  is  the  selling  price. 

Second  form.     Since  he  sells  at  a  loss  of  25%,  he  sells 
for  75  %  ,  or  |  of  the  cost. 


J  of  15/  = 
|  of  15/  =  3  times  -^-/  ==  1H/ 
.-.  etc. 


120 

Example  3.  A  hat  costing  $8  was  sold  for  $9, 
what  was  the  rate  of  gain  ? 

First  form.  (1.)  $9,  the  selling  price,  minus  $8,  the 
cost,  —  $1^  the  gain. 

(2.)  Wo  now  have  given  the  cost,  $8,  which  is  base, 
and  the  gain,  $1,  which  is  the  percentage,  to  find  the 
rate  of  gain,  which  is  rate  per  cent. 

Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent.,  the  rate  per  cent,  in  this  exam- 
ple is  found  by  dividing  $1  by  $8,  expressing  the  quo- 
tient as  hundredths.  The  quotient,  .12i,  is  the  re- 
quired rate  per  cent,  which  is  the  rate  of  gain.  .-.  etc. 

Second  form.    $9,  the  selling   price,   minus   $8,  the 

cost  price,  =  $1,  the  gain. 

$]  =  i  of  $8. 

i  of  a  number—  .12?  of  it. 

/.the  rate  of  gain  was  12J%. 

Example  4.  A  grocer  sold  coffee  at  8fi  above 
cost,  and  gained  20  per  cent. ;  required  the  cost. 

Form.  We  have  given  the  gain,  8/,  which  is  the 
percentage,  and  the  rate  of  gain,  .20;  which  is  rate  per 
cent.,  to  find  the  cost,  which  is  base. 

Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent.,  the  base  in  this  example  is  found 
by  dividing  8/  by  .20  ;  the  quotient,  40/,  is  the  re- 
quired base  which  is  the  cost.  .*.  etc. 

Example  5.  A  merchant  marked  goods  at  20% 
above  cost,  and  then  sold  at  20%  less  than  the  marked 
price.  Did  he  gain  or  lose  and  how  much  %  ? 

Form.     (1.)  Assume  $1  as  the  cost. 

(2.)  Since  he  marked  the  goods  at  20%  above  cost, 
the  marked  price  was  1.20  of  the  cost ;  and  we  have 


121 

given  the  cost,  81,  which  is  base,  and  the  rate  of  mark- 
ing, 1.20,  which  is  rate  per  cent.,  to  find  the  marked  price 
which  is  percentage.  [Solving  by  Case  I,  the  marked 
price  is  found  to  be  $1.20.] 

(3.)  Since  he  sold  at  20%  less  than  the  marked 
price,  he  sold  for  .80  of  the  marked  price,  and  we  have 
given  the  marked  price,  $1.20,  which  is  base,  and  the 
rate  of  selling,  .80,  which  is  rate  per  cent.,  to  find  the 
selling  price  which  is  percentage. 

Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent.,  the  percentage  in  this  example 
is  found  by  multiplying  $1.20  by  .80,  the  product,  $.96 
is  the  required  percentage,  which  is  the  selling  price. 

(4.)  The  cost,  $1,  minus  the   selling  price,  $.96  = 
$.04  —  the  loss. 

(5.)  We  now  have  given  the  cost,  $1,  which  is 
base,  and  the  loss,  $.04,  which  is  percentage,  to  find  the 
rate  of  loss,  which  is  rate  per  cent.  This  is  found  by 
dividing  $.04  by  $1,  expressing  the  quotient  as  hun- 
dredths.  The  quotient,  .04,  is  the  required  rate  per 
cent.,  which  is  rate  of  loss. 

Exercises. 

1.  14  barrels  of  flour  are  bought  at   $3.87£  each 
and  sold  at  $4.12?  each  ;  required  the  gain.      The  rate 
of  gain. 

2.  A  horse  is  bought  for  $94,  and  sold  at  4% 
gain ;  required  the  gain.     The  selling  price. 

3.  A  lot  is  bought  for  $2256  ;  for  what  must  it  be 
sold  to  gain  20  per  cent,? 

4.  There  was  a  gain  of  $14  realized  on  a  certain 
sale;  the  rate  of  gain  was  .15;  what  was  the  purchase 
price  ?     The  selling  price  ? 


122 

5.  A  grocer  bought  148  gallons  of  molasses  at 
26/  per  gal.,  and  sold  it  for  $64;  did  he  gain  or  lose? 
How  much  and  at  what  rate  ? 

6.  A  tarm  was  sold  for  $7400  which  was  5%  more 
than  it  was  worth;  required  its  value. 

7.  A  man  sold  two  horses   for   $120  each;  on  one 
he  gained  15%  and  on  the  other  he  lost  15%  ;  did  he 
gain  or  lose  on  the  entire  operation,  and  how  much  ? 

8.  I  sold  a  piece  of  property  for  $1000  gaining 
16%  ;  I  then  invested  the  $1000  in  another  piece  of 
property  which  I  was  forced  to  sell  at  a  loss  of  16%  ; 
did  I  gain  or  lose  on  the  series  of  transactions  and  at 
what  rate  ? 

9.  A  quantity  of  wheat  was  sold  for  $1248  which 
was  12  per  cent,  more  than  its  cost;  required  the  cost. 

10.  If  the  cost  aud  rate  of  gain  are  known  how 
find  the  gain  ?     The  selling  price  ?     Form  and  solve  a 
problem. 

11.  If  the  gain  and  the  rate  of  gain  or  loss  be  given 
what  can  be  found  and  how  ?  Form  and  solve  problems. 

12.  If  cost  and  selling  price  are  known  what  can 
be  found  and  how  ?     Form  and  solve  problems. 

13.  A  man  bought  corn  at  30/  per  bu.  and  sold  it 
at  a  gain  of  16f  %  ;  what  was  the  selling  price? 

14.  If  land  cost  $48  per  acre,   how   much   must  be 
asked  for  it,  that  a  10%  abatement  may  be  made  and  a 
profit  of  14%  still  be  realized  ? 

15.  A  merchant  asked  an   advance  of  35%,  but  af- 
terward sold  at  25%  less  than  his  asked  price;  did  he 
gain  or  lose  and  how  much  on  goods  that  cost  $72.25? 

16.  I  sold  f  of  an  article  for  what  the  entire  article 
cost ;  what  was  my  rate  of  gain  ? 


123 

17.  A  grocer  sold  a  hogshead  of  molasses  for  $30 
which  was  15%  more  than   it  cost;   required  the  cost 
per  gallon. 

18.  A  merchant  marks  a  piece  of  goods  $12,  but 
takes  off  6%  for  cash.     If  he  still   makes   10%  profit, 
what  was  the  cost  of  the  goods  ? 

19.  A  grocer  gains  14%   by   using   a  false  weight, 
required  the  weight  of  his  pound  weight. 

20.  A  man  bought  a  horse  for  $55  and  sold  him  for 
$70 ;  required  his  rate  of  gain. 

21.  If  f  of  an  article  be  sold  for   what  I  of  it  cost, 
what  is  the  rate  of  gain  ? 

22.  If  butter  be  bought   at  $24  per  cwt.,  for  what 
price  per  Ib.  must  it  be  sold  to   gain   15%,  and  allow  a 
discount  of  8%  for  cash  ? 

207.  Commission  and  Brokerage. 

Remark.    The  following  are  the  most  common  corresponding 
terms  involved. 

(1.)  The  sum  representing  the  amount  of  business 
transacted  is  base. 

(2.)  The  commission  or  brokerage  is  the  percentage. 
The  amount  involved,  or  the  sum  of  the  investment 
and  the  commission  is  the  percentage  in  some  instances. 

(3.)  The  rate  of  commission  or  brokerage  is  rate 
per  cent.  If  the  sum  of  the  investment  and  commis- 
sion be  the  percentage,  the  rate  per  cent,  is  j££  -f  the 
rate  of  commiRsion. 

208.  Forms  of  Solution. 

Example  1.  An  agent  sold  $1560  worth  of  goods 
and  charged 4%  for  his  services;  required  his  commis- 
sion. 


124 
/ 

Form.  We  have  given  the  sum  representing  the 
amount  of  business  done,  $1560,  which  is  base,  and  the 
rate  of  commission,  .04,  which  is  rate  per  cent.,  to  find 
the  commission,  which  is  the  percentage.  Since  the 
percentage  is  the  product  of  the  base  and  the  rate  per 
cent.,  the  percentage  in  this  example  is  found  by  mul- 
tiplying $1560  by  .04.  The  product,  $62.40,  is  the  per- 
centage, which  is  the  required  commission. 
[Give  other  forms  of  solution.] 

Example  2.  An  agent  was  paid  $24  for  buying 
$1440  worth  of  wheat ;  required  his  rate  of  commission. 

Form.  We  have  given  the  commission,  $24,  which 
is  percentage,  and  the  sum  representing  the  amount  of 
business  done,  $1440,  which  is  base,  to  find  the  rate  of 
commission  which  is  rate  per  cent. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent.,  the  rate  per  cent,  in  this  example  is 
found  by  dividing  $24  by  $1440,  making  the  quotient 
hundredths.  The  quotient,  .Olf  is  the  rate  per  cent, 
which  is  the  required  rate  of  commission.  .-.  etc. 
[Grive  other  forms  ot  solution.] 

Example  3.  A  banker  received  $40  for  making  a 
collection  at  4%;  required  the  sum  collected. 

Form.  We  have  given  the  commission,  $40,  which 
is  percentage,  and  the  rate  of  commission,  .04,  which  is 
rate  per  cent.,  to  find  the  sum  representing  the  amount 
of  business  done,  which  is  base. 

Since  the  percentage  is  the  product  of  the  base 
and  the  rate  per  cent.,  the  base  in  this  example  is 
found  by  dividing  $40  by  .04.  The  quotient,  $1000,  is 
the  base,  which  is  the  sum  representing  the  amount  of 
business  done.  .-.  etc. 

[Give  other  forms  of  solution.] 


125 

Example  4.  A  commission  merchant  received  $4160 
with  which  to  purchase  goods  after  deducting  his  com- 
mission of  4%;  required  the  sum  invested. 

First  form.  Since  the  rate  of  commission  was  .04, 
the  sum  received  was  1.04  of  the  sum  invested;  and 
we  have  given  the  amount  received,  $4160,which  is  per- 
centage, and  the  relation  of  this  sum  to  the  sum  invest- 
ed, 1.04,which  is  rate  per  cent.,  to  find  the  sum  invested 
which  is  base. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  percent.,  the  base  in  this  example  is  found  by 
dividing  $4160  by  1.04.  The  quotient,  $4000,  is  the 
base  which  is  the  required  sum  invested. 

Second  form.  Every  dollar's  worth  of  goods  pur- 
chased cost  the  sender  of  the  money  $1  for  the  goods 
-f-4/  for  agent's  commission,  or  $1.04;  hence  as  many 
dollars  were  spent  for  goods  as  $1.04  are  contained 
times  in  &4160,  or  4000.  .*.  the  sum  spent  for  goods 
was  $4000. 

Third  form.  Since  the  sum  invested  -f  4%  of  itself 
=  $4160, 104%  of  the  sum  invested  =  $4160, 
and  1%  of  the  sum  invested  =  Ti¥  of  $4160=  $40, 
"  lOOjgof  u       "  «       =  100  times  $40  ==  $4000. 

.*.  the  investment  was  $4000. 

[Exercises.] 

1.  At  3%  commission  what  does  an  agent  receive 
for  selling  $15360  worth  of  goods? 

2.  A  lawyer  collected  65%  of  a  debt  of  $348;  his 
rate  of  commission  was  4%;  what  did  he  receive? 

3.  A  merchant  sent  his  agent  $1250  with  Instruc- 
tions to  buy  goods;  how  much  was  expended  for  goods 
after  the  agent  deducted  his  commission  at 


126 

4.  How  many  barrels  of  flour  at  $4.50  per  barrel 
can  be  bought  for  $684,  if  3%  commission  be  deducted 
before  purchasing  ? 

5.  An  agent  received  $240  as  commission  at  4%; 
what  amount  of  business  did  he  transact? 

6.  The  rate  of  commission  was  5%;  the  sum  sent 
the  owner  as  proceeds  of  the  sale  was  $2275;  what  was 
the  commission? 

7.  What  amount  of  business  is  done  if  $36.50  is 
the  commission  at  2|%? 

8.  I  sent  my  agent  $516  to  invest  in  corn  after 
deducting  his  commission  at  3J%;  what  sum  was  paid 
for  corn  ? 

9.  An  agent  receives  a  remittance  of  $758  with 
which  to  buy  goods,  deduct  ing  his  commission  at  If  %\ 
what  is  his  commission? 

10.  If  the  merchant  for  whom  the  business  is  done 
(in  ex,  9)  send  check  for  the  commission,  what  is  its 
face? 

11.  A  consignment  of  grain  was  sold  for  $15650, 
of  which  $15540  were  net  proceeds  ;  required  the  rate 
of  commission. 

12.  An  agent  received  2%  commission  for  selling 
goods.     His   entire  commission  was  $126;  what  sum 
did  he  remit  his  employer? 

13.  I  sold  goods  on  commission  at  4%  through  a 
broker  who  charged  me  2-J%;  my  commission  after 
paying  the  broker  was  $216 ;  required  the  net  proceeds 
of  the  sales. 

14.  A  real  estate  agent  retains  as  commission  $124, 
sending  the  employer  $6575 ;  what  was  the  amount  of 
the  sale  and  the  rate  of  commission  ? 


127 

15.  An  agent  bought  25  horses  on  commission  at 
His  commission  was  $63  ;  required  the   cost  of 

each  horse. 

16.  If  I  pay  an  attorney  $48.50  for  making  a  col- 
lection at  5%;  what  was  the  claim? 

17.  How  many  barrels  of  flour  at  $5  each  can  an 
agent  buy  for  $324,  deducting  his  commission  of  3%? 

18.  An  agent  received   $3440  with  which   to  buy 
pork  after  deducting  his  commission  of  lj-%;  required 
the   number  of   pounds   of  por£  purchased  at  3/  per 
pound  ? 

19.  A  shipment  of  hay  sold  for  $14  per  ton  ;  the 
rate  of  commission  was  3%;   incidental  charges  $200; 
the  net  proceeds  were  $6290.     Required  the  number  of 
tons. 

209.  Stocks. 

1.  The  par  value  of  stock?  is  base.     Sometimes  the 
conditions   are  such   that  the  cost  or  another  number 
becomes  base. 

2.  The  premium,  the  discount,  or  the  cost,  is  per- 
centage. 

3.  The  rate  of  premium,  rate  of  discount,  rate  of 
purchase,  or  rate  of  sale  is  rate  per  cent. 

210.  Forms  of  Solution. 

.  Example  1.  A  company  pays  a  dividend  of  5%; 
what  does  that  man  receive  who  owns  12  shares? 

Form.  1'2  shares  of  $100  each  are  worth  $1200. 
We  now  have  given  the  par  value  of  the  stock,  $1200 
which  is  base  and  the  rate  of  premium,  .05,  which  is 
rate  per  cmt.,  to  find  the  premium,  which  is  percentage. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent.,  the  percentage  in  this  example  is 


128 

found  by  multiplying  $1200  by  .05.     The  product,  $60, 
is  the  percentage,  which  is    the    required   dividend. 
[Give  other  forms.] 

Example  2.  How  much  bank  stock  at  10%  dis- 
count, can  be  bought  for  $27945  ? 

First  form.  At  10%  discount,  the  stock  is  bought 
at  90%  of  the  par  value ;  and  we  have  given  the  pur- 
chase price,  $27945,  which  is  percentage,  and  the  rate 
of  purchase,  .90,  which  is  rate  per  cent.,  to  find  the  par 
value,  which  is  base. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent., the  base  in  this  example  is  found  by 
dividing  $27945  by  .90.  The  quotient,  $31050,  is  the 
base,  which  is  the  required  par  value, 

Second  form.  Since  the  stock  is  bought  at  10%  dis- 
count, every  $1  worth  of  stock  is  bought  for  $.90.  As 
many  dollars'  worth  of  stock  can  be  bought  for  $27945 
as  $.90  is  contained  times  in  $27945,which  equal  31050. 
.-.  $31050  worth  of  stock  can  be  bought  for  $27945  at 
10%  discount. 
[Give  other  forms.] 

Example  3.  Stock  was  bought  at  120  and  sold  at 
128;  what  was  the  rate  of  gain? 

Form.  At  120,  $1  worth  of  stock  was  bought  for 
$1.20,  and  at  128  it  was  sold  for  $1.28.  The  gain  was 
$.08.  We  have  given  the  cost  of  the  stock,  $1.20, 
which  in  this  case  is  base,  and  the  gain,  $.08,  which  is 
percentage.  Since  the  percentage  is  the  product  of 
the  base  and  the  rate  per  cent.,  the  rate  per  cent,  in 
this  example  is  found  by  dividing  $.08  by  $1.20,  ex- 
pressing the  quotient  as  hundredths.  The  quotient, 
.06f,  is  the  rate  percent.,  which  is  the  required  rate  of 
gain. 


129 

Example  4.  A  broker  bought  stock  at  3%  discount 
and  sold  at  3%  premium,  and  gained  $420;  required  the 
face  of  the  stock  bought. 

Form.  Since  he  bought  at  3%  of  the  face  less  than 
the  face  and  sold  for  3%  of  the  face  more  than  the  face, 
he  gained  6  %  of  the  face,  but  $420  equaled  his  gain; 
hence  6%  of  the  face  of  the  stock  equaled  $420,  then 
\%  of  the  face  ==  \  of  $420  =  $70,  and  100%  of  the  face 
=  100  times  $70  =  $7000.     .-.  etc. 
[Give  other  forms.] 

Exercises. 

1.  What  cost  60  shares  railroad  stock  at  4%  pre- 
mium? 

2.  Midland  stock  bought  at  3%  discount  was  sold 
at  4%  premium ;  required  the  gain  on  75  shares. 

3.  9  shares  of  I.  &  St.  L.  stock  at  96  are  exchanged 
for  mining  stock  at  104;  how  many  $100  shares  of 
mining  stock  are  purchased? 

4.  The  net  earnings  of  a  street  railway  are  $1500 ; 
the  capital  invested  is  $30000;  required  the  rate  of 
dividend  that  can  be  declared. 

5.  How  many  shares  of  canal  stock  at  94  can  be 
bought  for  $118800? 

6.  At   20%    premium   how  many  shares  of  stock 
will  0800  buy? 

7.  If  Vandalia  stock  is  quoted  at  102J  how  much 
stock  can  be  bought  for  $3246,  brokerage  being  \%*l 

8.  Pan  Hancfle  stock  bought  at  110  and  sold  at  116 
yields  what  rate  of  gain  ? 

9.  Wm.  Smith  receives  $630   as  a  7%  dividend; 
how  manv  $50  shares  does  he  own  ? 


130 

10.  A  man  owned  25  shares  of  rolling  mill  stock  of 
$50   each;    the   company  declared  a  dividend  of  8%, 
payable  in   stock:  how  many   additional   shares  were 
issued  to  him  ? 

11.  A  bridge  company  whose'stock  was  $12500  re- 
quired an  assessment  of  $625;  what  rate  did  they  de- 
clare ? 

12.  Mr.  Wells  owns  35  shares  of  $100  each  in  a 
turnpike  company;  his  dividend  was  $201.25;  required 
the  rate  of  dividend. 

211.  Insurance. 

1.  The  sum  insured  is  base. 

2.  The  premium  or  the  sum  insured  ±  the  premi- 
um is  a  percentage  of  the  sum  insured. 

3.  The  rate  of  premium  or  |££  ±  the  rate  of  pre- 
mium is  the  rate  per  cent.' 

Remark.  If  the  premium  is  viewed  as  the  percentage,  the 
rate  of  premium  is  rate  per  cent.;  but  if  the  amount  or  differ- 
ence be  viewed  as  percentage,  the  rate  per  cent,  is  yW  — 
the  rate  of  premium. 

212.  Forms  of  Solution. 

.  Examples.  1.  A  house  is  insured  for  $800  at  4%; 
required  the  premium. 

2.  The  premium  paid  for  insuring  a  barn  for  $1200 
is  $6;  required  the  rate  of  insurance. 

3.  A  man   paid  $150  for  insuring  goods  at  3  per 
cent.;  required  the  value  insured. 

Remark.  The  above  problems  come  within  Cases  I,  II,  and 
III,  respectively,  and  are  readily  solved  by  forms  similar  to 
those  given  under  the  preceding  applications  of  percentage. 


131 

4.  A  stock  of  goods  worth  $19600  is  insured  a 

so  as  to  recover  both  the  value  of  the  goods  and  the 
premium  in  case  of  loss;  required  the  sum  insured. 

first  form.  Since  the  premium  is  2%  of  the  sum 
insured,  the  value  of  the  goods,  $19600,  must  be  98%  of 
the  sum  insured.  We  have  given  $19600,  the  value  of 
the  goods,  which  is  percentage,  and  .98,  the  rate  of  dif- 
ference, which  is  rate  per  cent.,  to  find  the  sum  insured, 
which  is  base,  Since  the  percentage  is  the  product,  &c. 

Second  form.     Since  2%  of  the  sum  insured  =  the 
premium,  98%  of  the  sum  insured  must  =the  value  of 
the  property  =  $19600  ;  then  l%of  the  sum  insured  = 
Jg  of  $19600  =  $200,  and  100%  of  the  sum  insured  = 
100  times  $200  =  $20000.     .-.  etc. 

5.  A  shipper  took  out  a  policy  for  $34200  to  include 
the  value  of  the  goods  shipped  and  also  the  premium 
at  6  per  cent.;  required  the  value  of  the  goods. 

Form.  Since  the  premium  was  6%  of  the  face  of 
the  policy,  the  value  of  the  goods  must  have  been  94% 
of  its  face.  We  have  given  the  sum  insured,  $34200, 
which  is  base,  and  the  rate  of  difference,  .94,  which  is 
rate  per  cent,  to  find  the  value  of  the  goods,  which  is 
percentage. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent.,  etc. 
[Give  other  forms.] 

Exercises. 


1.  Property  is   insured  for  $2250  at  3£%;  what  is 
the  premium  ? 

2.  A  stock  of  goods  is  insured  for  $4800  at  §%; 
what  is  the  premium  ? 


132 

3.  A  man   pays  $24  for  insuring  a  house  at  -J  per 
cent.;  what  is  the  face  of  the  policy? 

4.  The  rate  of  insuring  a  house  was  \\%  and  the 
premium  was  $12  ;  what  was  the  face  of  the  policy? 

5.  $56  is  the  premium  paid  on  a  policy  of  $11200  ; 
what  is  the  rate  ? 

6.  A  shipment  of  grain  worth  $15600  is  insured  at 
2%,  so  as  to  include  the  premium  as  well  as  the  value 
of  the  grain  in  case  of  loss ;    what  is  the  face  of   the 
policy? 

7.  The  rate  of  premium  is  3^%  and  the  value  of 
property  insured  is  $2488;  what  face  of  policy  will  in- 
clude both  in  case  of  loss  ? 

8.  To  include  the  premium  at  \\%  and  the  value 
ol  the  goods,  the  face  of  a  policy  is  $6300;  what  is  the 
value  of  the  goods  ? 

9.  A  vessel's  cargo  is  insured  for  $17600;  the  pre- 
mium at   4%  is   included;  what   is   the  value  of  the 
cargo  ? 

10.  What  will  it  cost  to  insure  a  house  for  $900  at 
\\  per  cent.?     At  f  per  cent.? 

11.  A  piece  of  property  was  insured  for  $5000 ;  the 
premium  paid  was  $30 ;  what  was  the  rate  ? 

12.  At  2i%  what  sum  must  be  insured  on  property 
worth  $3648  to  include  both  property  and  premium  in 
case  of  loss  ? 

13.  A  merchant  insured  a   consignment  of  goods 
for  $13728  so  as  to  include  both  property  and  premium; 
required  the  premium. 

14.  If  a  man  pays  $38.80  for  insuring  f  of  the  value 
of  his  house,  what  is  its  value  if  the  rate  of  insurance 
is 


133 

15.  A  building  is  insured  so  as  to  include  f  of  its 
value  and  the  entiro  premium.  The  value  of  the  build- 
ing is  $24500,  and  the  rate  of  insurance  \\%\  what  is 
the  premium? 

213.     Taxes. 

1.  The    assessed   value   of  the   property  taxed   is 
base. 

2.  The  rate  of  taxation  is  rate  per  cent. 

3.  The  tax  is  the  percentage. 

Remark.    Use  forms  of   solution  similar  to  those  already 
given. 

Exercises. 

1.  What  sum  must  be  assessed  in  order  that  $12500 
may  remain  after  paying  a  commission  of  4%  for  col- 
lection ? 

2.  The  valuation  of  the  property  in  a  certain  dis- 
triet  is  $2364748.     A  tax  of  12560  is  required.     What 
tax  must  that  man  pay  whose  property  is  assessed  at 
$8000? 

3.  If  the  rate  of  tax  was  $12  on  $1000  and  the  tax 
levied  was  $14674,  what  was  the  valuation? 

4.  Required  A's  tax  on  property  worth  $2460  at 


5.  What  sum  must   be  assessed  to  raise  a  net  tax 
of  $7400,  and  pay  a  commission  of  2%  for  collection? 

6.  In  a  school  district  a  school  is  maintained  by  a 
tax  on  the  property  of  the  district,  which  is  valued  at 
$648750.  A  teacher  is  paid  $45  per  month  for  6  months, 
and   other   expenses    are   $64.50  ;  required   the  tax  on 
property  worth  $4865. 


134 

7.  Find  A's  tax  from  the  following  items  : 

His  district  paid  in  teachers'  salaries,     .       $1200.00 
"  "     for  fuel,     ......          57.60 

"  "      "   incidentals,     ....        38.00 

The  money  received  from  the  school  fund  was  $258. 
The  remaining  expense  was  paid  by  a  rate  bill.  The 
aggregate  attendance  was  9568  days,  while  A  sent  4 
pupils  46  days  each. 

8.  The  cost  of  a  public  worl^was  $1260.    The  rate 
of  taxation  was  3  mills  on  the  dollar,  and  the  collect- 
or's commission  was  3-;  what  was  the  valuation? 


214.     Customs,  or  Duties. 

1.  The   net  quantity  -of  goods   is  the  base  in  com- 
puting specific  duties. 

2.  The  cost  of  the  goods  in  the   country  whence 
they  were  exported  is  the  base  in  computing  Ad  Valo- 
orem  duties. 

Remark.    Duties  are  assessed  upon  the  goods  actually  im- 
ported.    All  deductions  are  made  previous  to  the  assessment. 

3.  The  rate  of  duty  is  rate  per  cent. 

4.  The  duties  assessed  constitute  the  percentage. 

Remark.    Use  forms  of   solution   similar  to  those  already 
given. 

Exercises. 

^ 

1.  An  importation  was  56  casks  of  wine,  each  con- 
taining 36  gallons;     The  net  duty  at  30%  Ad  Valorem, 
•amounted  to  $907.20  ;  required  the  invoice  per  gallon. 

2.  A  quantity  of  lace  was  invoiced  at  $816.54.  The 
merchant  paid  the  duties  and   a  freight  bill  of  $22.50, 
and  found  that  the  total  cost  was  $980.50  ;  required  the 
rate  of  duty. 


135 

3.  Required  the  duty  at  2?  cents  per  pound  on  2700 
pounds  ol  cloves,  tare  being  b%. 

4.  If  the  duty  on  opium  is  100%,  required  the  im- 
port tax  on  236  Ib.  of  opium  invoiced  at  $3.75  per  Ib. 

5.  Required  the  Ad  Valorem  duty  at  30%  on   125 
boxes  of  tea,  each  containing  70  Ib.  and  invoiced  at  85/ 
per  Ib.,  tare  being  8  Ib.  per  box. 


Applications  of  the  Second  Class. 
215.     Interest. 

1.  The  principal  is  base. 

2.  The  rate  of  interest  is  rate  per  cent.  .  The  rate 


of  the  amount  or  of  the  difference,  i.  e.  i^-g-  ±  the  rate 
of  interest  may  be  rate  per  cent. 

3.  The  interest  is  percentage.     The  amount  is  also 
a  percentage  of  the  principal. 

4.  The  number  representing  the  time  in  years  is  a 
factor  used  with  the  principal  and.  the  rate  of  interest 
to  determine  the  interest. 

Remarks.  1.  The  time  unit  in  interest  is  1  year  of  12  months 
of  30  days  each. 

2.  Interest  has  involved  in  it  four  terms,  viz  :  —  the  principal, 
the  rate  of  interest,  the  time  in  years,  and  the  interest.  The 
first  three  of  .these  are  factors  of  the  fourth. 

216.     CASE  1. 

Given  the  principal,  the   rate  of  interest  and  the 
time  to  find  the  interest. 

Example.     Required  the  interest  of  £500  for  2  yr. 
3  mo.  15  da.  at  6.  - 


136 

First  form.     2yr.  3  mo.  15  da.  —  2^7¥  yr. 

Since  the  interest  is  the  product  of  the  principal, 
the  rate  of  interest  and  the  number  representing  the 
time  in  years,  the  interest  in  this  example  is  found  by 
multiplying  together  $500,  .06  and  2^.  The  product, 
$68.75,  is  the  required  interest. 

Remark.    The  following  expressions  for  the  above  form  in- 
dicate a  method  for  finding  interest  for  months  or  days: 

For  months.  \  Principal  X  rate X  time  in 
1  12 


-ei      ,  f  PrincipalX^ateXtime  in  days      T   , 

jror  days.     •< — ?—=  interest. 

Second  form. 

Int.  of  $500  for  1  yr.  at  6%  —  $30.00 

<•    500     «  1  mo.  "  6%  =  TL  of    $30  =  2.50 

"    500    "  1  da.   "  6%  =-=  ¥L  of  $2.50—  .08J- 

"  '<  500  U2yr.    "  6%  =  2  times  $30  =  $60.00 

"  "  500  "3  mo.  "   6^=3     "    $2.50—  7.50 

"  "  500  "  15  da.  "  6%  =  15  "        .08J—  1.25 

"  "  500  u  2  yr.  3  mo.  15  da.  at  6%  =  $68.75 

Exercises. 

1.  The  principal  is  $150,  the  rate  of  interest  6%, 
the  time  3  yr.;  required  the  interest. 

2.  Find  the  interest  of  $750  tor  2  yr.  8  mo.  at  5%. 

3.  Find  the  interest  of  $6750  for  5  yr.  10   mo.  15 
da.  at  8%. 

4.  Find  the  interest  ot  $75  for  11  mo.  at  9%. 

5.  Find  the  interest  of  $956  for  7  mo,  at  10%. 

6.  Find  the  interest  of  $1278  for  5  mo.  3  da.  at  6 
per  cent. 


137 


7.  Find  the  interest  of  $16  for  90  da.  at 

8.  Find  the  interest  of  $800  for  5  yr.  4  mo.  20  da. 
at  10  per  cent. 

9.  Find  the  interest  of  $1750  for  6  yr.  2  mo.  28  da. 
at  8  per  cent. 

10.  Find  the  interest  of  $9.50  for  20  da.  at  10%. 

11.  Find  the  interest  of  $17850  for  40  da.  at  11%. 

12.  Find   the   interest  of    $675  from  July  3,  1881, 
to  August  7,  1883,  at  8  per  cent. 

13.  Find  the  interest  of  $95  from  May  10,  1880,  to 
April  6,  1884,  at  7  per  cent. 

14.  A  note  was  drawn  for  $850  on  January  8,  1882; 
a    payment   of  $200  was  made  September     18,    1882; 
what  was  due   January  8,  1883,  rate  of  interest  being 
5  per   cent  ? 

15.  A  note  was  drawn  for  $1000  on  June  6,  1878  ; 
its  rate  of  interest  was  6%.     A  payment   of  $450  was 
made   June   6,1879;    another   of  $200,  September  21, 
1879;  another   of   $350,  May  10,    1880;  what  was  due 
December  25,  1881  ? 

16.  A  note  was  drawn  for  $67  on  October  17,  1882; 
the  rate  of  interest  being  9%.     A  payment  of  $16  was 
made   December    1,   1882;   another   of  $22,  March    12, 
1883;    another   of  fc]8,  June  11,  1883;  another  of  $7, 
September  17,  1883;  what  was  due  October  17,  1883  ? 

17.  What  is  the   compound   interest  of  $156  for  4 
yr.  6  mo.  at  6  per  cent.? 

18.  What  is  the  compound  interest  of  $1350  for  3 
yr.  8  mo.  at  6  per  cent.? 

19.  What  is  the  compound  interest  of  $1100  for  1 
yr,  6  mo.  at  8%;  interest  compounded  quarterly? 


138 

20.  What  is  the   compound   interest  of  $800   for  3 
yr,  at  5%;  interest  compounded  semi-annually? 

21.  What  is  the  annual  interest  of  $750  for  3  yr.  4 
mo.  at  6  per  cent.? 

22.  Find  the  annual  interest  accruing  on  $65  for  5 
yr.  8  mo.  12  da.  at  7  per  cent. 

217.     CASE  II. 

Given,  the  principal,  the  time  and  the  interest,  to 
find  the  rate  of  interest. 

Example.  The  principal  is  $500,  the  time  2-^  yr. 
and  the  interest  $68.75 ;  required  the  rate. 

First  form.  Since  the  interest  is  the  product  of 
the  principal,  the  rate  and  the  number  representing  the 
time  in  years,  the  rate  of  interest  in  this  example  is 
found  by  dividing  $68.75  by  the  product  of  $500  and 
2^4  and  making  the  quotient  hundredths.  The  quo- 
tient, .06,  is  the  required  rate, 

Second  form.  The  interest  of  $500  for  2^¥yr.  at  \% 
=  $11J^.  Now  since  $11^  =  tne  interest  of  $500  for 
2^  yr.  at  1%,  $68.75  —  the  interest  of  $500  for  227¥ 
yr.  at  as  many  times  \%  as  68.75  is  times  llji  which 
=  6.  6  times  \%  =-  6%.  .'.  the  required  rate  of  in- 
terest is  .06. 

Exercises. 

1.  The  principal  is  $380,  the  time  3  yr.  4  mo.  and 
the  interest  $76.76  ;  required  the  rate, 

2.  A  man   received  $218.40  interest  on   a  loan   of 
$780  for  4  yr.  8  mo.;  required  the  rate. 

3.  The  principal  is  $4760,  the  time  2  yr.  8  mo.  20 
da.  the  interest,  $864 ;  required  the  rate. 

4.  The  interest  is  $456.84,  the  time  5  yr.  9  mo.  15 
da.;  what  is  the  rate,  the  principal  being  $986  ? 


139 

5.  The  principal  was  $960,  the  time  7  yr.  5  mo.  and 
the  interest  $520.80;  required  the  rate. 

6.  The  principal  is  $480,  the  time  6  yr.  3  mb.  and 
the  interest  $210;  what  is  the  rate  ? 

7.  The   time   is   8  yr.   9  mo.  12  da.,  the   interest 
$17.46,  and  the  principal  $26.50  ;  required  the  rate. 

8.  The  principal  is  $2015,  the  interest  $575.40  and 
the  time  5  yr.  8  mo.  15  da.;  required  the  rate. 

218.     CASE  III. 

.  Given  the  principal,  the  interest  and  the  rate,  to 
find  the  time. 

Example.     In  what  time  will  $500  yield  $68.75  at 


First  form.  Since  the  interest  is  the  product  of 
the  principal,  the  rate  and  the  number  representing 
the  time  in  years,  the  time  in  this  example  is  found  by 
dividing  $68.75  by  the  product  of  $500  and  .06.  The 
quotient.  2-^-;  is  the  number  representing  the  time  in 
years.  ,  /.  the  time  is  2-/¥  yr.  or  2  yr.  3  mo.  15  da. 

Second  form.  Since  $500  at  interest  for  1  yr.  at 
6%  yield  $30,  $500  must  be  on  interest  for  as  many 
times  1  yr.  at  6%  to  yield  $68.75  as  $68.75  are  times 
$30,  which=22^.  2^  time  1  yr.=2^  yr,=2  yr.  3  mo. 
15  da,  .*.  etc. 

Exercises. 

1.  The   principal  is   $4080,  the  interest  $668.10, 
and  the  rate  5%;  required  the  time. 

2.  The  principal  is  $176,  the  interest  $22  ;  and  the 
rate  7%;  required  the  time. 

3.  The  principal  is  $1300,  the  interest  $274  and 
the  rate  8%;  required  the  time. 


140 

4.  How  long  will  it  take  any  principal  to  double 
itself  at 4jg?  at 5%? at  6%?  at  10$? 

219.     CASE  IV. 

Given  the  interest  or  amount,  the  time,  and  the 
rate  of  interest,  to  find  the  principal. 

Example.  What  principal  will  yield  $68.75  int.  in 
2  yr.  3  mo.  15  da.  at  6^? 

First  form.  Since  the  interest  is  the  product  of 
the  principal,  the  rate  of  interest,  and  the  number  rep- 
resenting the  time  in  years,  the  principal  in  this  ex- 
ample is  found  by  dividing  $68.75  by  the  product  of 
.06  and  2^.  The  quotient,  $500,  is  the  required  prin- 
cipal. 

Second  form.  -A  principal  of  $1  will  yield  $.13| 
interest  in  2^  yr.  at  6%;  and  to  yield  $68.75  interest 
in  the  same  time  at  the  same  rate  would  require  a 
principal  as  many  times  SI  as  $68.75  are  times  $.131, 
which  =  500.  500  times  $1  =  $500.  /.  etc. 

Exercises. 

1.  Required   the  principal  if  the  time   is  3  yr.  8 
mo.,  the  rate  .06,  and  the  interest  $462. 

2.  Required   the   principal  if  the   time  is  6  yr.  3 
mo.,  the  rate  .07,  and  the  interest  £64.26. 

3.  What   principal   will   in   7  yr.  4  mo.  at   8% 
amount  to  $749.70? 

4.  What  ?um  will  yield  $185  int.  in  18  mo.  at  5%? 

5.  What  sum  must  be  invested   in  6%  stocks  to 
yield  an  income  of  $1500? 

6.  What  principal  will  amount  to  $200  in  14  yr. 
3f  mo.  at7%? 

7.  What  principal  will  amount  to  $355.60  in  "2  yr. 
7  mo.  at 


141 


Review  Exercises  in  Interest. 


Principal. 

Rate. 

Time. 

Interest.     Amount. 

1. 

$420 

6 

% 

3 

yr. 

6 

mo. 

9 

? 

2. 

49215 

6 

it 

1 

yr. 

3 

ino. 

18 

da. 

? 

? 

3. 

? 

6 

u 

4 

yr. 

$24 

? 

4. 

300 

6 

u 

9 

36 

9 

5. 

9 

6 

a 

1 

yr. 

6 

mo. 

72 

9 

6. 

9 

8 

a 

2 

yr. 

11 

mo 

.27 

da. 

$845 

7. 

12.80 

7 

u 

3 

yr. 

4 

mo. 

3  da.            ?              ? 

8. 

? 

5 

tl 

2 

yr. 

4 

mo. 

12 

da. 

40 

9. 

750 

4 

a 

9 

120 

10. 

3542 

? 

u 

9 

yr. 

6 

mo. 

15 

da. 

11. 

? 

9 

" 

9 

yr. 

9 

mo 

900 

12. 

1475 

10 

a 

5 

yr. 

mo.  5 

da. 

9 

13. 

50 

8 

u 

9 

20 

14. 

500 

? 

« 

2 

yr- 

(5 

mo. 

50 

15. 

?   • 

6 

a 

3 

yr. 

2 

mo. 

5 

16. 

9 

6 

" 

2 

yr. 

6 

mo. 

76 

17. 

1000  . 

*4 

a 

1 

yr. 

8 

mo. 

? 

18. 

9 

5 

" 

30 

da. 

2072 

19. 

530 

6 

u 

8 

yr. 

2 

mo. 

21 

da. 

? 

9 

20. 

176.25 

9 

a 

1 

yr. 

11 

mo 

.5 

da. 

25.52 

• 

21. 

185.85 

3J 

u 

3 

vr. 

5 

mo. 

15 

da. 

? 

22. 

? 

12 

U 

'90 

da. 

412 

23, 

6 

it. 

2 

yr 

6 

mo. 

9 

690 

24. 

7 

" 

6 

yr. 

157.50 

? 

25. 

? 

8 

u 

i 

yr. 

6 

mo. 

24 

da. 

30.24 

26. 

82.50 

6 

a 

5 

yr. 

8 

mo. 

12 

da. 

? 

27. 

450 

? 

a 

1 

yr. 

8 

mo. 

12 

da. 

61.20 

28. 

600 

9 

" 

? 

798 

29. 

375 

8 

u 

? 

90 

30. 

? 

5 

a 

3 

yr. 

341.25 

Bt 

400 

7 

U 

? 

68.60 

32. 

700 

9 

u 

? 

924.70 

142 

220.     Discount. 

Discount  is  treated  under  three  heads,  viz:  True 
Discount,  Bank  Discount  and  Commercial  Discount. 

a.  True  Discount.  True  Discount  is  an  application 
of  Case  IV  in  interest. 

1.  In  true  discount,  the  present  worth  is  the  principal. 

2.  The  debt  is  amount,  and  may  be  considered  as 
percentage  if  the  time  element  be  reduced  to  1  yr. 

Remark.  The  rate  of  interest  and  the  number  representing 
the  time  in  years  being  factors,  if  one  of  them  be  divided  and 
the  other  multiplied  by  the  same  number,  the  value  of  the  pro- 
duct is  not  affected,  e.  g  If  the  time  should  be  3  yr.  and  the 
rate  .06,  we  may  reduce  the  time  element  to  1  yr.  by  dividing 
it  by  3  if  at  the  same  time  we  multiply  the  rate,  .06,  by  three. 

It  is  evident  that  the  interest  of  a  gh  en  principal  for  3  yr. 
at  6  per  cent,  equals  the  interest  of  the  same  principal  for  1  yr. 
at  18  per  cent. 

3.  The  discount  is  the  percentage.  (.Interest). 

4.  The  rate   of  interest  is  rate  per  cent.     The   rela- 
tion (expressed  in  hundredths)  of  the  debt  to  the  pres- 
ent worth  may  be  the  rate  per  cent. 

5.  If  the  debt  be  considered  as  percentage,  the  rate 
per  cent,  becomes  ±$$  -f  the  given  rate  multiplied  by 
the  number  representing  the  time  in  years ;    the  time 
element  being  divided  by  itself  and  thus  reduced  to  1 
year. 

6.  The  time  named  is  the  time  element,  except  as 
stated  in  2  and  5. 

Example.  Required  the  present  worth  and  true 
discount  of  $568.75  due  in  2  yr.  3  mo.  15  da.  if  money 
is  worth  6. 


143 

First  form.  The  time  element  is  reduced  to  1  yr. 
by  dividing  it  by  2^¥.  By  multiplying  the  rate,  .06, 
by  2^-,  it  becomes  .13J. 

If  the  debt,  $568.75,  be  considered  as  percentage,  the 
rate  per  cent,  becomes  1.131. 

We  thus  have  given  the  percentage,  $568.75,  and 
the  rate  per  cent.,  1.131.,  to  find  the  base,  which  is  the 
required  present  worth. 

Since  the  percentage  is  the  product  of  the  base  and 
the  rate  per  cent.,  the  base  in  this  example  is  found  by 
dividing  $568.75  by  1.13|.  The  quotient,  $500,  is  the 
required  base,  or  present  worth. 

Second  form.  $1  at  interest  for  2  yr.  3  mo.  15  da.  at 
6%,  amounts  to  $1.13f.  $1  is,  therefore,  the  present 
worth  of  $1.13f  due  in  2  yr.  3  mo.  15  da.  without  inter- 
est, money  being  worth  6%.  Hence  the  present  worth 
of  $568.75  is  as  many  times  $1  as  $568.75  are  times 
$1.131,  which  =  500.  500  times  $1  =  $500.  .  .  etc. 

Exercises. 

1.  Required  the  present  worth  and  true  discount 
of  $436  due  3  years  hence,  if  money  is  worth  12%. 

2.  A  note  of  $4800  is  due  in  4  years.     What  is  its 
cash  value  if  money  is  worth  5  %  per  annum  ? 

3.  A  was  offered  a  lot  for  $225  cash  or  $230  in  3  mo. 
Did  he  make  or  lose,  and  how  much  by  accepting  the 
latter  offer  ? 

4.  Required  the  discount  of  a  debt  of  $864  due  in 
8  mo.  if  paid  now  ? 

5.  A  grocer   bought   62  barrels  of  molasses  of  31| 
gal.  each,  at   26/  per  gal.,  on   90  days'  time,  and  sold 
immediately  for  $615  ;  how  much  did  he  gain  if  money 
was  worth  8%? 


144 

6.  What  is  the  present  worth  and  true  discount  of 
$27.50  due  in  20  months,  if  money  is  worth  6%? 

7.  Hogs  were   purchased  to  the  value  of    $1574, 
one-half  payable  in  3  mo.  and  the  remainder  in  6  mo., 
without  interest.    What  is  the  cash  value  of  the  stock 
if  money  is  worth  7%? 

8.  Which  is  worth  the  most  $640  in  12  mo.,  $620 
in  6  mo.  or  $600  in  cash,  if  money  is  worth  8%? 

6.  Bank  Discount  Bank  Discount  is  the  simple 
interest  of  a  given  sum  for  the  time  elapsing  between 
the  date  of  discounting  and  the  date  of  legal  maturity. 
This  time  is  called  the  term  of  discount. 

1.  The  face  of  the  obligation  is  principal  or  base. 

2.  The  rate  of  discount,  or  rate  of  proceeds  is  rate 
per  cent. 

3.  The  discount  or  proceeds  is  percentage. 

4.  The  term  of  discount  is  the  time  element. 

Remarks.  I.  If  an  interest  bearing  note  be  discounted  at  a 
bank,  the  amount  of  the  note  is  found  at  the  given  rate  of  in- 
terest for  the  time  elapsing  between  the  date  of  the  note  and  its 
legal  maturity.  The  amount  thus  found  is  then  discounted  at 
the  rate  of  discount  for  the  term  of  discount. 

2.  If  the  face  of  the  note,  (principal),  the  rate  of  discount, 
(rate  of  interest),  and  the  term  of  discount  (the  time)  be  given 
to  find  the  discount,  (interest),  the  problem  is  solved  under 
Case  I  in  interest. 

3.  If  the  face  of  a  note,  (principal),  rate  of  discount,  (rate  of 
interest),  and  term  of  discount,  (time  element),  be  given  to 
find  the  proceeds,  the  discount,  which  is  interest,  is  found  un- 
der Case  I  in  interest.     The  discount  is  then  subtracted  from 
the  face  of  the  note ;  the  remainder  being  the  required  pro- 
ceeds ;  or  the  rate  of  proceeds  may  be   used  as  rate  per  cent, 
and  the  problem  solved  directly  by  Case  I  in  interest. 

4.  If  the  proceeds,  rate  of  discount  or  rate  of  proceeds,  and 
term  of  discount  be  given  to  find  the  face  of  the  note,  the 
problem  is  solved  under  Case  IV  in  interest,  considering  the 
proceeds  as  interest,  and  the  rate  of  proceeds  as  rate  of  interest, 
first  reducing  the  time  element  to  1,  and  increasing  the  given 
rate  of  discount  in  the  same  ratio. 


145 

Example.  Given  the  proceeds  $493,  the  term  of 
discount,  63  days,  and  the  rate  of  discount,  .08,  to  find 
the  face  of  the  note. 

Solution.  The  time  element,  -££$,  divided  by  itself 
is  reduced  to  1. 

The  given  rate  of  discount,  .08,  multiplied  by 


The  rate  of  proceeds  is,  therefore,  ,98f  . 

The  following  form  exhibits  the  work. 

$493 


1X.986 


=  $500  =  the  face  of  the  note. 


[NOTE.— For  practical  purposes  the  form  of  solution  usually  given  in  the 
text  hooks  for  the  ahove  problem  is  preferable  to  the  solution  here  given.j 

Exercises. 

1.  Find  the  bank  discount  and  proceeds  of  a  note 
of  $80  payable  in  60  days,  discounted  at  8%? 

2.  A  note  of  $56  dated  Jan.  1,  1884,  and  payable 
May  1,  1884,  is  discounted  in  bank   at  6%;  required 
the  proceeds. 

3.  A  note  of  $500,  dated  Dec.  15, 1883,  and  payable 
Feb.  18,  1884,  is  discounted   at   9%;  required  the  pro- 
ceeds and  the  discount. 

4.  A  note  of  $650  with   interest  at  8%  is  dated 
Nov.   30,  1883,  and   is  payable  in  90  days.    It  is  dis- 
counted Jan.  5,  1884,  at  10%;  required  the  proceeds. 

5.  A  note  of  $1400  with  interest  at  10%  is  dated 
Jan.  16,  1884,  and  is  payable  May  18,  1884.     It  is  dis- 
counted April  7,  1884,  at  12%;    required  the  proceeds. 

6.  For  what  sum  must  a  60  days  note  be  drawn 
that  when  discounted  in  bank  at  6%  the  proceeds  may 
be  $1000? 


146 

7.  A  owes  B  $1500 ;  for  what   sum  must  a  90  days 
note   be  drawn  that  when  discounted  in  bank  at  6%, 
B  may  obtain  his  money? 

8.  A  merchant  bought  goods  for  $1621.20  cash,  ob- 
taining the  money  from  a  bank  on  a  60  days  1%  note  ; 
what  was  the  face  of  the  note  ? 

9.  Required  the  cash  value  of  a  note  of  $6780  dis- 
counted in  bank  for  4  mo.  15  da.  at  6%? 

10.  Required  the  face  of  a  note  given  in  bank  that 
when  discounted  for  5  mo.  21  da.  at  7%,  the  proceeds 
shall  be  $57-97? 

c.  Commercial  Discount.  Commercial  Discount  is 
a  deduction  from  the  face  of  a  bill  or  other  obligation 
without  regard  to  time;  It  is  also  called  per  cent,  off, 
and  is  effected  under  Case  I  in  Percentage. 

Exercises, 

1.  A  merchant  bought  $1200  worth  of  goods  on  6 
mo.  time  ;  but  paid  cash  on  obtaining  a  discount  of  8% 
off.     What  did  he  pay? 

2.  6%  off  was  allowed  en  a  bill  of  goods  amount- 
ing to  $1878.50 ;  what  sum  did  they  cost? 

3.  A  country  merchant  purchased  a  bill  of 'goods 
amounting  to  $3075  on  4  mo.;  but  was  offered  5%  off 
for  cash.     Would  he  gain  by  borrowing  the  money  from 
a  bank  at  8%  per  annum  for  the  time  and  paying  the 
cash? 

4.  What  is  the  cash  value  of  goods  listed  at  $5650, 
10%  off  for  wholesale  and  5%  off  for  cash  ? 

5.  A  merchant  paid  $1.14  per  yd.  for  goods  after  a 
discount  of  6%  had  been  made  from  the  marked  price. 
What  was  the  marked  price  ? 


147 

6.  What  was  the  invoice  price  of  goods  for  which 
I  paid  $89  after  a  discount  of  40%  had  been  made? 

221.     Exchange. 

1.  The  face  of  a  draft  is  base. 

2.  The  exchange  or  the  cost  of  a  draft  is  percent- 
age. 

3.  The  rate  of  exchange  or  the  rate  of  cost  is  rate 
per  cent.     The  rate  of  cost  is  also  called   the  course  of 
exchange. 

4.  In  time  drafts  the  time  named  plus  3  days  is 
the  time  element. 

5.  The  interest  accruing  on  a  time  draft  is  deduct- 
ed from  the  face  of  the  draft,  for,  the  hank  having  the 
use  of  the  money  during  the   time  should,  in  equity  ^ 
pay  the  interest. 

6.  Make  each  problem  an  application  of  either  the 
first  or  the  second  class  of  the  applications  of  percent- 
age, according  as  the   element  of  time  is   or  is  not  in. 
volved. 

Exercises. 

1.  Required  the   cost  of  a  draft   on  New  York  at 
\%  premium. 

2.  What  is   the  cost  of  a  draft  on   Chicago  at  f  % 
discount? 

3.  Required  the  cost  of  a  draft  for  $560  payable  30 
days  after  sight,  exchange  \%  premium,   and  inter- 
est 6%. 

4.  What  is  the  face   of  a  30  days  draft  which  cost 
$352.62,  exchange   being  \.\%  discount,  and  interest  6 
per  cent.? 


148 

5.  A  New  York  merchant  sold  $1284   worth   of 
goods.     Would  he  better  draw  on  the  purchaser,  pay- 
ing \\%  f°r  collection,  or  require  a  sight  draft  on  New 
York  for  the  bill,  exchange  being  2£%  premium? 

6.  What  is  the  face  of  a  draft  on  Indianapolis  at 
45  days,  exchange  being  at  a  premium  of  3%? 

7.  How  much  must  I  pay  in  Paris  for  a  draft  on 
Chicago  for  $4500  at  18f  /  per  franc? 

8.  What  must  be  paid  in  New  York  for  a  draft  on 
London  of  560£  at  8%  premium  ? 

9.  What  is  the  face  of  a  60  days  draft  that  when 
sold  will  yield  $1000,  exchange  \%  discount,  and  in- 
terest 6^? 

10,  Eequired  the  cost  of  a  30  days  draft  for  $1920 
at  f  %  discount,  interest  7%. 

11.  What  is  the  face  of  a  60  days  draft  that  can  be 
bought  for  $3195.20,  interest  8%  and  exchange  \\% 
premium. 

222.     Equation  of  Payments. 

Remark.  A  problem  in  equation  of  payments  usually  con- 
tains a  number  of  problems  under  Case  I  in  interest;  the  ob- 
ject being  to  find  an  equitable  time  for  the  payment  of  several 
sums  of  money  due  at  different  times. 

Exercises. 
Remark.    For  methods  of  solution  see  text  books. 

1.  Required  th^  average  term  of  credit  for  the  fol- 
lowing debts:  $400  due  in  3  mo.,  $500  due  in  5  mo.  and 
$700  due  in  8  mo. 

2.  A  debt  of  $2400  is  subject  to  the  following  con- 
ditions: $800  is  due  in  4  mo.,  $600  is  due  in  6  mo.,  and 
the  remainder  is  due  in  8  mo.     What  is  the  average 
term  of  credit  ? 


149 

3.  A  man  owes  $240  due  in  20  days,  and  $560  due 
'in  30  days.     At   the  end  of  16  days  he  pays  $300  and 

at  the  end  ot  24  days  he  pays  $350;  when,  in  equity, 
should  he  pay  the  remainder? 

4.  A  merchant  bought   goods  as  follows :  April  1, 
$280  on  3  mo.  time,  $200  on  4  mo.,  $300  on  5  mo.,  and 
$560  on  6  mo.     On  what  da,te  will   a  single  payment 
discharge  the  debts? 

5.  Wm  Smith  owes  $30  due  in  60  days,     $100  due 
in  120  days,  and  $750  due   in  180  days ;  what  is  the 
equated  time  of  payment  ? 

6.  Mr.  Wallace  bought  grain  on  a  credit  of  90  days 
to  the  following  amounts  : 

25th  of  Jan.        .     .    .      $3750 

10th  of  Feb 3000 

6th  of  March      ....   2400 

On  the  first  day  of  May   he  wishes  to  give  his  note  for 
the  amount.     At  what  time  will  it  mature? 

7.  A  merchant  bought  goods  as  follows :    Feb.  10, 
$364;  March  12,  $375;  April  15,  $554;  May  18,  $622.  He 
obtains  6   months' credit   on  each    purchase ;  at  what 
time  can  the  whole  be  equitably  discharged? 

8.  What  is  the  average  date  for  paying  three  bills, 
due  as  follows;  Jan.  31,  $477;  Feb.  28,  $377;  March  31, 
$777? 

9.  Kequired  the  average  time  of  the  following 
bills,  allowing  to  each  term  of  credit  3  days  of  grace. 
— April  3,  $500  on  3  mo.;  April  4,  $200  on  2  mo.;  April 
4,  $200  cash  ;  and  April  10,  $500  on  3  months. 

10.  Find  the  equated  time  for  the  payment  of  the 
following  notes  :  $350  dated  July  12,  1883,  for  60  days  ; 
$720  dated  fc*ept.  lt»,  1883,  for  90  days  ;  and  $1200  dated 
Nov.  5,  for  120  davs. 


150 

11.  A  owes  $600  due  in  6  months,  but  at  the  end  of 
3  months  he  desires   to   make   a   payment   sufficiently' 
large  that  the   remainder   may  not  be   payable  until  6 
months  after  its  first  date  of  maturity ;  how  large  must 
be  the  payment  ? 

12.  On  tho  first  day  of  Jan.    B   takes  a  house  at  a 
rental  of  $300   per  annum,    agreeing   to   pay   the  rent 
quarterly  in  advance;  required  the  equated  time  for  the 
payment  of  the  whole. 


SECTION    XIII. 

RATIO  AND  PROPORTION. 

Ratio- 

223.  Ratio  is  the  relation  of  one  number  to  another 
considered  as  a  measure. 

Examples.  The  ratio  of  6  to  3  =  2  ;  i.  e.  if  3  be  ap- 
plied as  a  measure  to  6,  the  number  of  applications  that 
can  be  made  is  2.  2  is,  therefore,  the  relation  that  6 
sustains  to  3  considered  as  a  measure. 

The  ratio  of  1  to  2  =  £  ;  of  2  to  3  =t,  etc. 

Remarks.  1.  "The  number  of  times  that  a  divisor  is  contained 
in  a  dividend  is  the  ratio  of  the  dividend  to  the  divisor.  A 
quotient  if  abstract,  may  be  viewed  as  a  ratio 

A  multiplier  ia  the  ratio  of  the  product  to  the  multiplicand. 

2.  The  part  that  one  number  is  of  another  is  called  the  ratio 
of  the  first  to  the  second.     Ratio  is  thus  related  to  fractions  in 
that  it  is  the  relation  of  a  part  to  a  whole. 

3.  Since  ratio  is  the  relation  of  measure,  the  two  numbers  be- 
tween which  a  ratio  exists  are  like  numbers. 

224.  The  Terms  Used.  Three  terms  are  concerned 
in  thinking  a  ratio  :  viz, — a  dividend,  a  divisor,  and  a 
quotient.  These  are  called,  respectively,  Antecedent, 
Consequent  and  the  Eatio. 


151 

Antecedent.  The  dividend,  or  first  term  of  a  ratio 
is  called  the  antecedent* 

Consequent.  The  divisor,  or  second  term  of  a  ratio 
is  called  the  consequent. 

The  Ratio.  The  quotient  of  the  antecedent  by  the 
consequent  is  called  the  ratio. 

225.  The  Notation  of  a  Ratio. 

A  colon  is  used  to  separate  the  written  terms  of  a 
ratio,  thus:  6:3.  This  expression  is  read  the  ratio  0* 
6  to  3  ;  it  expresses  the  quotient  of  6  by  3. 

226.  Classes. 

Simple.  A  ratio  each  of  whose  terms  is  a  single 
number  either  integral,  fractional  or  mixed  is  called  a 
simple  ratio.  As  6  :  2;  I >  :  f ;  2i  :  3J. 

Compound,  Two  or  more  simple  ratios,  viewed 
together  as  to  the  product  of  their  corresponding 
terms  constitute  a  compound  ratio. 

227.  Principles. 

Remark.  Since  the  terms  antecedent,  consequent  and  ratio, 
signify  dividend,  divisor  and  quotient,  respectively,  the  gener- 
al principles  of  division  become,  by  a  change  in  terminology, 
the  principles  of  ratio. 

I.  Multiplying  the  antecedent  multiplies  the  ratio, 

II.  Multiplying  the  consequent    divides   the  ratio. 

III.  Multiplying  both    antecedent  and  consequent 
by  the  same  number  does  not  change  the  ratio. 

IV.  Dividing  the  antecedent  divides  the  ratio. 

V.  Dividing  the   consequent    multiplies  the  ratio. 


152 

VI.  Dividing  both  antecedent   and  consequent  by 
the  same  number  does  not  change  the  ratio, 

Remark.  Prin.  Ill  may  be  thus  stated  :  The  ratio  between 
like  multiples  of  two  numbers  equals  the  ratio  between  the 
two  numbers.  Prin.  VI  may  be  thus  stated  :  The  ratio  be- 
tween like  parts  of  two  numbers  equals  the  ratio  between  the 
two  numbers. 


Proportion. 

228.  Two  equal  ratios  constitute  a  proportion. 

Remark.  The  ratios  which  form  a  proportion  may  both  be 
simple,  both  compound,  or  one  simple  and  the  other  com- 
pound. 

A  Simple  Proportion.  A  proportion  that  consists 
of  two  simple  ratios  is  called  a  simple  proportion. 

A  Compound  Proportion.  A  proportion  containing 
a  compound  ratio  is  called  a  compound  proportion. 

229.  Notation. 

A  proportion  is  notated  by  writing  a  double  colon 
between  the  two  equal  ratios  and  interpreting  it  by 
the  word  as.  The  proportion,  2:4 ::  5: 10,  is  read  2  is 
to  4  as  5  is  to  10.  A  proportion  is  often  notated  by 
writing  the  two  equal  ratios  with  the  sign  of  equality 
between  them.  The  proportion,  2:3  =  4:6,  is  read— 
The  ratio  oi  2  to  3  equals  the  ratio  of  4  to  6. 

Remark.  The  antecedent  of  the  first  ratio  and  the  conse- 
quent of  the  second  ratio  of  a  proportion  are  called  the  extreme 
terms  and  the  other  terms  the  mean  terms  of  the  proportion. 

230.  Principle. 

The  product  of  the  mean  terms  of  any  proportion 
equals  the  product  of  its  extreme  terms. 

Demonstration.  In  a  ratio  the  antecedent  is  the 
product  of  two  factors,  viz.:  the  consequent  and  the  ra- 


153 

tio.  In  a  proportion  the  product  of  the  mean  terms  is 
composed  of  three  factors,  viz.:  The  ratio  and  the 
consequent  of  the  second  ratio  (which  compose  the  an- 
tecedent of  the  second  ratio,)  and  the  consequent  of  the 
first  ratio. 

The  product  of  the  extreme  terms  is  composed  of 
ihree  factors,  viz.:  the  ratio  and  the  consequent  of  the 
first  ratio  (which  compose  the  antecedent  of  the  first 
ratio)  and  the  consequent  of  the  second  ratio. 

It  ib  thus  observed  that  the  factors  composing  the 
product  of  the  means  are  identical  with  the  factors 
composing  the  product  of  the  extremes  ;  hence  the  two 
products  are  equal. 

231.    Forms  of   Solution. 

Remarks.     1.  Any  problem  that  is  solvable  by  proportion  is 
readily  solved  by  analysis. 

2.  It  is  to  be  remembered  that  a  ratio  exists  between  like 
numbers  only. 

3.  In  solving  a  problem  by  proportion  two  distinct  steps  are 
taken. 

a.  The  arrangement  of  the  ratios,  called  the  statement  of  the 
proportion. 

b.  The  reduction  of  the  proportion,  or  the  finding  of  the  un- 
known term. 

Example  1.  If  11  bu.  of  wheat   cost  $9,  what  cost 
17  bu.  ? 
Written  form.  Thought  form. 

9:x::  11:17  |  Since  a  ratio  exists  between  like 
numbers  only,  a  ratio  exists,  in  this  example,  between 
11  bu.  and  17  bu.,  and  between  $9,  the  cost  of  11  bu., 
and  the  cost  of  17  bu.,  and  these  ratios  must  be  equal. 
9,  which  represents  the  cost  of  11  bu.,  may  be  made 
either  term  of  either  ratio,  and  the  required  number, 
which  we  will  represent  by  x,  will  be  the  other  term 
of  the  same  ratio.  We  choose  to  make  9  the  antece 


154 

dent  of  the  first  ratio ;  then  is  x  the  consequent  oi  the 
first  ratio.  $9  is  the  cost  of  11  bu.,  while  $#  is  the  cost 
of  17  bu.,  a  greater  number  than  11,  hence  the  conse- 
quent, x,  of  the  first  ratio  is  greater  than  its  antece- 
dent, 9 ;  and  since  the  two  ratios  must  be  equal  to 
form  a  proportion,  the  consequent  of  the  second  ratio 
must  be  greater  than  its  antecedent.  Hence  we  make 
11  (bu.)  the  antecedent  and  17  (bu.)  the  consequent  of 
the  second  ratio. 

We  now  have  the  two  extremes  and  one  mean  of 
a  proportion.  Since  the  product  of  the  extremes  equals 
the  product  of  the  means,  the  required  mean  is  found 
by  dividing  the  product  of  the  extremes  by  the"  given 
mean,  (using  the  numbers  abstractly  in  performing 
the  operation.)  The  required  mean  is  13.91.  .'.  17 
bu.  cost  $13.91. 

Example  2.  If  75  men  can  build  a  wall  50  ft.  long, 
8  ft.  high  and  3  ft.  thick,  in  ten  days,  how  long  will  it 
take  100  men  to  build  a  wall  150  ft.  long,  10  ft.  high 
and  4  ft.  thick  ? 

Written  form.  Thought  form, 

x:  10::  75: 100  I      In  this  problem  a  ratio    exists  be- 


150:50 

10:8 

4:3 


tween   75  men   and  100  men ;    50  ft. 
length   and  150  ft.  length  ;  8  ft.  hight 

and  10  ft.  hight;  3  ft.  thickness  and  4  ft.  thickness  ;  10 
days  and  the  required  number  of  days  which  we  may 
represent  by  x. 

10  (days)  may  be  made  either  term  of  either  ratio, 
and  x  (days)  will  be  the  other  term  of  the  same  ratio. 
We  will  make  10  the  consequent  of  the  first  ratio,  then 
will  x  be  the  antecedent  of  the  first  ratio. 

10  days  are  required  for  75  men  to  do  a  work  while 
x  days  are  required  for  100  men  to  do  the  work.  100 


155 

men  require  a  less  number  of  days  than  75  men,  hence 
the  antecedent,  #,  of  the  first  ratio  is  less  than  its  con- 
sequent, 10 ;  and  since  the  two  ratios  must  be  equal  to 
form  a  proportion,  the  antecedent  of  the  second  ratio 
is  less  than  its  consequent ;  we  therefore,  write  75  as 
the  antecedent  and  100  as  the  consequent  of  the  sec- 
ond ratio. 

Again, — 10  days  are  required  to  build  a  wall  50 
feet  long,  while  x  days  are  required  to  build  a  wall  150 
feet  long,  a  greater  length  than  50  feet,  hence  x  is 
greater  than  ten,  and  as  the  two  ratios  must  be  equal 
to  form  a  proportion,  the  antecedent  of  the  second  ra- 
tio must  be  greater  than  its  consequent ;  we  therefore 
write  150  as  the  antecedent  and  50  as  the  consequent 
of  the  second  ratio. 

Again, — 10  days  are  required  to  build  a  wall  8  ft. 
high,  while  x  days  are  required  to  build  a  similar  wall 
10  ft.  high,  a  greater  hight  than  8  ft.,  hence  x  is  more 
than  ten,  and  since  the  two  ratios  must  be  equal  to  form 
a  proportion,  the  antecedent  of  the  second  ratio  is 
greater  than  its  consequent ;  we  therefore  write  10  as 
the  antecedent  and  8  as  the  consequent  of  the  second 
ratio. 

Again, — 10  days  are  required  to  build  a  wall  3  feet 
thick,  while  x  days  are  required  to  build  a  similar  wall 
4  ft.  thick,  a  greater  thickness  than  3  ft.,  hence  x  is 
more  than  10,  and,  since  the  two  ratios  must  be  equal 
to  form  a  proportion,  the  antecedent  of  the  second  ratio 
is  greater  than  its  consequent;  we  therefore,  write  4  as 
the  antecedent  and  3  as  the  consequent  of  the  second 
ratio. 

We  now  have  given  the  factors  of  the  means  and 
the  factors  of  one  extreme  of  a  compound  proportion  to 
find  the  other  extreme. 


156 

Since  the  product  of  the  extremes  equals  the  pro- 
duct of  the  means,  the  required  extreme  is  found  by 
dividing  the  product  of  the  factors  of  the  means  by  the 
product  of  the  factors  of  the  given  extreme.  [The  work 
of  reducing  the  proportion  may  be  shortened  by  con* 
sidering  the  factors  of  the  means  as  factors  of  a  divi- 
dend and  the  factors  of  the  given  extreme  as  factors  of 
a  divisor,  and  canceling  common  factors.]  The  re- 
quired extreme  is  found  to  be  37£.  .'.  the  required  time 
is  37J  days. 

Exercises. 

1.  If  70  horses  cost  $3500,    what  cost  160  horses? 

2.  If  5  Ib.  of  coifee  cost  $1.35,  what  cost  9  Ib.  ? 

3.  If  12  tons  of  hay  cost  $87,  what  cost  17  tons  ? 

4.  If  a  tarm  cost  $4800,  what  cost  f  of  it  ? 

5.  If  i  yd.  of  silk  cost  $1.65,  what  cost  8  yd.? 

6.  If  19  acres  of  land  sell   for  $570,  required  the 
price  of  40  acres. 

7.  How  many  men  will  do   as  much   work  in  84 
days  as  8  men  do  in  126  days? 

8.  If  I  borrow  $3500  for  30   days,   for  what  time 
may  I  return  $900  to  requite  the  favor? 

9.  If  a  10  cent  loaf  weigh  s   1  Ib.    2  oz.    when  flour 
is  $7^  per  barrel,  what  should  it  weigh  when  flour  is  $6 
per  barrel  ? 

10.  What  cost  147  bu.  of  corn  if  35  bu.  cost  $28J? 

11.  If  15  sheep  cost   $15.90,    how   many  sheep  can 
be  bought  for  $155.82? 

13.  At  36  d.  per  $£,    what   is   1£   worth  in  U.  S. 
money  ? 

14,  In  what  time  can   a   man   pump   54  barrels  of 
water,  if  he  pump  24  barrels  in  1  hr.  14  min  ? 


157 

15.  If  10  bales  of  cotton   can   be  carried  115  miles 
for  $8,  how  far  can    15  bales   be   carried    for  the  same 
money  ? 

16.  If  $1200,  in  2  yr.  3    mo.   gain  $162  interest  at 
6%,  how  much  will  $800  gain  in  3  yr.  3  mo.  at  8%  ? 

17.  A  mill  makes  1265  barrels  of  flour  by  running 
10  hr.  per  day  for  13  days  ;    how   many   barrels  would 
it  make  in  26  days,  running  15  hours  per  day  ? 

18.  If  it  cost  $28  to  carpet   a  room   12  ft.  by  7  ft., 
what  will  it  cost  to  carpet  a  room  30  ft.  by  18  ft.  ? 

19.  If  60  men  dig  a  canal  80  rd.  long,  12  ft.  wide 
and  6  ft.  deep  in  18  days,  working  16  hr.  per  day,  how 
many  men  will  be  required  to  dig  a   ditch  30  rd.  long, 
9  ft.  wide  and  4  ft.  deep  in  24  days,  working  12  hours 
per  day  ? 

10.  A  min  gained  $3155  on  the  sale  of  91  horses  ; 
how  much  would  he  have  gained  on  245  horses  sold  at 
the  same  rate  ? 

21.  If  a  tank  17^  ft.  long,    11|   ft.  wide  and  13  ft. 
deep,  contain  546  barrels,  how  many  barrels  will  a  tank 
hold  that  is  16  ft.  long,  15  ft.  wide  and  7  ft.  deep  ? 

22.  If  the  interest  $346  for  1  yr.  8  mo.  is  $32,  what 
is  the  interest  of  $215.25  for  3  yr.  4  mo.  24  da.  at  the 

same  rate  ? 

23.  A  certain  bin  is  8  ft.  by  4£  ft.  by  2*  ft.  and 
its  capacity  is  75  bu. ;    how  deep  must  a  bin  be  to  con- 
tain 450  bu.,  if  it  be  28  ft.  long  by  3J  ft.  wide? 

24.  If  366  men  in  5  days  of  10  hr,  each,  can  dig  a 
trench   70  yards  long,  3  yards  wide,  and  2  yards  deep ; 
what  length  of  trench  5  yards  wide  and  3  yards  deep, 
can  240  men  dig  in  9  days  of  12  hr.  each? 


158 

25.  If  it  cost  $64  to  pave  a  walk  3  ft.  wide  and  32  ft. 
long,  what  will  it  cost  to  pave  a  walk  5  ft.  wide  and  64 
ft.  long? 

26.  If  7  men,  working  10  hr.  per  da.,  make  6  wag- 
ons in  21  days,  how  many  wagons  can  12  men  make  in 
16  days  working  9  hours  per  day  ? 

27.  If  $8  yield  $2  interest  in  3  mo.   12  da,  at  9%, 
how  much  int.  will  $140  yield  in  9  mo.  22  da.  at  6%  ? 

28.  If  it  cost  $320  to  buy  the  provisions  consumed 
in  8  mo.  by  a   family  of  7  persons,  how  much   at  the 
same  rate  will  it  cost  to  feed  a  family  of  twice  the  num- 
ber of  persons  for  i  as  much  time? 

29.  If  a  stone  2  ft.  long  10  in.  wide,  and  8  in.  thick, 
weigh  72  lb.,  required  the  weight  of  a  similar  stone  6  ft- 
long,  15  in.  wide  and  6  in.  thick. 

30.  If  300  lb.  of  wool  at  28/  per  lb.  are  exchanged 
for  36  yd.  of  cloth  1J  yd.  wide,  how  many  lb.  of  wool  at 
35/  per  lb.,  should   be  given  for  20  yd.  of  cloth  f  yd. 
wide? 


Proportional  Parts. 

232.  A  number  is  divided  into  proportional  parts  if 
separated  into  parts  whose  ratio  equals  the  ratio  of 
given  numbers. 

Example.  Divide  40  into  two  parts  that  are  to 
each  other  as  3  to  5. 

40  is  to  be  divided  into  parts  that  are  respectively, 
equi-multiples  of  3  and  5;  i.e.,  one  of  the  required 
parts  of  40  is  as  many  times  3  as  the  other  part  is 
times  5. 


159 

40,  the  sum  of  the  required  parts  is,  therefore,  the 
same  number  of  times  the  sum  of  3  and  5.  40  is  5 
times  the  sum  of  3  and  5.  5  times  3  =  15  and  5  times 
5  =  25.  Hence  15  and  25  are  the  required  parts  into 
which  40  is  to  be  divided. 

Principles,  a,  The  sum  of  the  proportionals  is  to 
the  sum  of  the  required  numbers  as  the  less  propor- 
tional is  to  the  less  of  the  required  numbers. 

b.  The  sum  of  the  proportionals  is  to  the  sum  of 
the  required  numbers  as  the  greater  proportional  is  to 
the  greater  of  the  required  numbers. 


SECTION  XIV. 
INVOLUTION  AND  EVOLUTION. 

233.    Table  for  Determining  the   Number  of  Terms 
(Orders)  in  the  Square  Root  of  a  Given  Square  Number. 

12=1  From  the  marginal  table  it  is  seen 

93=81  that  a  square  consisting  of  one  or  two 

102— 100  terms  contains  one  term  in  its  square 

992=9801  root ;  a  square  of  three  or  four  terms 

100I=10000  contains  two  terms  in  its  square  root; 

9992— 998001  a  square  of  five  or  six  terms  contains 

10002=1000000  three  terms  in  its  square  root,  etc. 
99992=99980001 

.12=.01  The  principle  may  be  formulated 

,9^=.81  thus : 

.Ol2— .0001  If  a  square  number  expressed  in  the 

.992=:.9801  decimal  scale  be  separated  into  periods 

.0012=.000001  of  two  terms  each,  beginning  at  the  inter- 

.9992=.998001  val  between  units  and  tenths,  there  are  as 

etc.  many  terms  in  its  square  root  as  there  are 

periods  in  the  square. 


160 


234.  Table  Indicating  the  Lowest  Place  which  a  Sig- 
nificant Figure  can  Occupy  in  the  written  Product  of  the 
Given  Factors. 

Remark.  The  following  abbreviations  are  used  — u  for  units, 
t  for  tens,  h  for  hundreds,  th  for  thousands,  tth  for  ten 
thousands,  .t  for  tenths,  .h  lor  hundredth*,  .th  for  thousandths, 
.tth  for  ten  thousandths,  etc. 


ua=  u. 

t  X  u  =  t. 

u  X  .t    =  .t 

tf=h. 

h  X  u  =  h. 

u  X  .h   =  .h 

h2=  tth. 

th  X  u  =  th. 

u  X  .th  =  ,th 

h2=m. 

h  X  t  =  th. 

.t  X  .h  =  .th 

.t2—  .h 

th  X  t  =  tth. 

.t  X  .th  =  .tth 

h2=  .tth 

etc. 

etc. 

etc. 

235.     A  Number  Consisting  of  Tens  and  Units  Involved 
to  the  Second  Power. 


Example.     M*=  what  ? 

Remark.  In  the  light  of  the  definition  of  a  second  power,  34 
is  squared  by  multiplying  it  by  itself.  In  order  that  the  par- 
tial products  be  readily  seen,  they  should  not  be  combined  by 
addition  until  they  are  all  found. 

The  involution  may  be  shown  in  the  form  that  follows. 


=  t.X  u  ) 

=  t  X  u) 


34 
34 

16"=  42  =  u2. 
120  =  30  X  4    =t.X 

gW=W  =  t2. 

1156  =  342  =  t2  +  2t  X  u  +  u2. 

Remark.  If  any  number  consisting  of  tens  and  units  be 
squared,  it  is  readily  seen  that  the  same  steps  will  be  taken  as 
in  the  above  example,  i.  e.,  the  square  of  a  number  consisting 
of  tens  and  units  is  composed  of  the  square  of  the  tens  plus 
twice  the  tens  multiplied  by  the  units  plus  the  square  of  the 
units . 


.161 

236.  A  Number  Consisting  of  Hundreds,  Tens  and 
Units  Involved  to  the  Second  Power. 

Example.     234!  =  what  ? 
234 
234 

16  =  42  =  u2. 
120  =  30  X  4  =  t  X  u. 
800  =  200  X  4  =  h  X  u. 
120  =  4_X  30  =  t  X  u. 
900  =  30*  =  t2, 
6000  =  200  X  30  =  h  X  t. 
800  =  4  X  200  =  h  X  u. 
6000  =  30  X  200  =  h  X  t. 
40000  ==  2002  =  h2. 

54756  =  234*  =_  h2  +  2hXt  -ft2-f  2[h+t]Xu+u2. 

Remark.  If  any  number  consisting  of  h,  t  and  u  be  squared 
the  same  steps  will  be  taken  as  in  the  above  example ;  i.  e., 
the  square  of  a  number  consisting  of  h,  t  and  u  is  composed  of 
the  square  of  the  h  plus  twice  the  h  multiplied  by  the  t  plus 
the  square  of  the  t  plus  twice  the  sum  of  the  h  and  t  multi- 
plied by  the  u  plus  the  square  of  the  u. 

237.  A  Number  Consisting  of  Tenths  and  Hundredths 
Involved  to  the  Second  Power. 

Example.     .24*  =  what  ? 

,24 
.24 

.0016  =  .042  =  .h2. 
.008    =.2  X  .04=.t  X  ,h.. 
.008    ==  .04  X  -2  =  -t  X  -h. 
.04      =  .22  =  .t2. 


.0576  =  .242  =  .t2+2.tX.h+.h2. 

Remark.  It  is  thus  seen  that  the  square  of  a  number  con- 
sisting of  tenths  and  hundredths  is  composed  of  the  square  of 
the  tenths  plus  twice  the  tenths  multiplied  by  the  hundredths 
plus  the  square  of  the  hundredths. 


132  * 

238.  A  Number  Consisting  of   Tenths,  Hundredths 
and  Thousandths  Involved  to  the  Second  Power. 

Example.     .246*  =  what  ? 

.246 

.246 

.000036  =  .006*  =  .th2. 
.00024  =  .04  X  .006=  .h  X  -th. 
.0012   =  .2  X  -006  =  .t  X  -th. 
.00024  =  .006  X  .04  =  .h  X  .th. 
.0016  =  .042  =  .h2. 
.008   =  .2  X  .04  =-=  .t  X  -h. 
.0012   =.006  X  .2=.t  X  .th. 
.008   =  .04  x  .2  =  .t  X  -h. 

.04    ==  .22  =  .t2. 

.060516  =  .2462=.t2+2.tX.h+.h2+2[.t+.h]X.th-hth2. 

Remark.  It  is  thus  seen  that  the  square  of  a  number  con- 
sisting of  tenths,  hundredths  and  thousandths  is  composed  of  the 
square  of  the  tenths  plus  twice  the  tenths  multiplied  by  the 
hundredths,  plus  the  square  of  the  hundredths,  plus  twice  the 
sum  of  the  tenths  and  the  hundredths  multiplied  by  the  thou- 
sandths, plus  the  square  of  the  thousandths. 

239.  A  Number  Consisting  of  Tens,  Units  and  Tenths 
Involved  to  the  Second  Power. 


Example,    32.62  =  what  ? 
32.6 
32.6 

.36  =  .62  =  .t2. 
1.2    =  2  X  .6  :=  u  X  .t. 
18.      =  30  X  .6  =  t  X  .t. 
1.2    =  .6  X  2  =  u  X  -t. 
4.      =  22  =  u2. 
60.      =  30  X  2  =  t  X  u. 
18.      =  .6  X  30  =  t  X  -t. 
60.      =2_X30  =  tXu. 
900.      =  302  =  t2. 
1062.76  =32762= 


163 

Remark.  It  is  thus  seen  that  the  square  of  a  number  con- 
sisting of  tens,  units  and  tenths  is  composed  of  the  square  of  the 
tens,  plus  twice  the  tens  multiplied  by  the  units,  plus  the  square 
of  the  units,  plus  twice  the  sum  of  the  tens  and  the  units  mul- 
tiplied by  the  tenths,  plus  the  square  of  the  tenths. 

240.  General  Formula  Embodying  the  Involution  of 
any  Number  in  the  Decimal  Scale  to  the  Second  Power. 

A  general  principle  for  exhibiting  the  elements 
which  compose  the  second  power  of  a  number  consist- 
ing of  any  number  of  terms  (orders)  in  the  decimal 
scale  may  be  thus  stated  :— 

The  square  of  a  number  consisting  of  any  number 
of  terms  is  composed  of  the  square  of  the  first,  or  high- 
est term  plus  twice  the  first  term  multiplied  by  the  sec- 
ond, plus  the  square  of  the  second,  plus  twice  the  sum 
of  the  first  two  terms  multiplied  by  the  third,  plus  the 
square  of  the  third,  plus  twice  the  sum  of  the  first 
three  terms  multiplied  by  the  fourth,  plus  the  square 
of  the  fourth,  plus  twice  the  sum  of  the  first  four  terms 
multiplied  by  the  fifth,  plus  the  square  of  the  fifth, 
plus,  etc. 

Remark.  A  fraction  is  involved  to  the  second  power  by 
squaring  both  its  terms. 

241.  Evolution  of  the  Second  Root. 

Example  L    i/TISS  =  what  ? 

Written  form.  Thought  form. 

1156(34 


900 
6t)256 
240 


16 
16 


Since  there  are  four  terms  in  the  square 
there  are  two  terms  in  its  square  root,viz. — 
tens  and  units. 

The  square  of  a  number  consisting  of  t 
and  u  is  composed  of  the  t2-|-2tXu-j-u2. 
The  sq.  of  t  =  h ;  hence  the  h  of  the  power  con- 
tain  the  sq.  of  the  t  of  the  root.    The  greatest  square 


164 

number  of  h  in  11  h  is  9  h,  the  sq.  root  of  which  is  3  t. 
9  h  taken  from  the  power  leave  256.  This  remainder 
contains  a  product  of  which  twice  the  tens  of  the  root, 
or  6  t,  is  one  factor  and  the  units  of  the  root  is  the 
other  factor.  Since  tXu  =  t,  the  25  t  of  the  remain- 
ing part  of  the  power  contain  the  required  product. 
Since  25  t  contain  a  product  of  which  one  factor 
is  6  t,  tLe  other  factor  is  found  by  dividing  25  t  b}T  6  t; 
the  quotient,  4,  is  supposed  to  be  the  units  of  the  root. 
The  product  of  6  t  by  4  =  24  t,  which  taken  from  the 
remaining  part  of  the  power  leave  16.  This  remain- 
der must  contain  the  square  of  the  u  of  the  root.  The 
square  of  4  =  16,  which  taken  from  the  remaining 
part  of  the  power  leave  nothing.  1156  is  thus  found 
to  be  a  square  number ^of  which  34  is  the  square  root. 


Example  2.     1/54756  =  what  ? 
Written  form.  Thought  form. 

54756(234 
4 


Since   there  are  five  terms  in  the 


given  square  there  are  three  terms  in 
its  square  root,  viz. — h,  t  and  u. 

The  square  of  a  number  consisting 
of  h,  t  and  u,  is  composed  of  the 
square  of  the  h,  plus  twice  the  h  mul- 
tiplied by  the  t,  plus  the  square  of  the 
t,  plus  twice  the  sum  of  the  h  and  the 
t  multiplied  by  the  u,  plus  the  square 
of  the  u. 

The  sq.  of  h  —  tth ;  hence  the  tth  of  the  power- 
contain  the  sq.  of  the  h  of  the  root.  The  greatest 
square  number  of  tth  in  5tth  is  4tth ;  the  square  root 
of  which  is  2h.  4tth  taken  from  the  power  leave 
14756.  This  remainder  contains  the  product  of  two 
factors,  one  of  which  is  twice  the  h  of  the  root,  or  4  h, 


165 

and  the  other  the  t  of  the  root.  Since  hxt  =  th,  the 
th  ol  the  remaining  part  of  the  power  contain  the  re- 
quired product  ....  Since  14  th  contain  a  product  of 
which  one  factor  is  4  h,  the  other  factor  is  found  by  di- 
viding 14  th  by  4  h,  the  quotient,  3  t,  is  supposed  to  be 
the  tens  of  the  root.  The  product  of  4h  by  3t  =  12  th, 
which,  taken  from  the  remaining  part  of  the  power 
leave  2756.  This  remainder  contains  the  sq.  of  the  t 
of  the  root.  The  square  of  3t  =  9h, which  taken  from 
the  remaining  part  of  the  power  leave  1856.  This 
remainder  contains  the  product  of  two  factors,  one  of 
which  is  twi«;e  the  sum  of  the  h  and  the  t  of  the 
root,  or  46  tens,  and  the  other  the  units  of  the  root. 

Since  t  X  u  =  t,  the  tens  of  the  remaining  part  of 
the  power  contain  the  required  product. 

Since  185  tens  contain  a  product  of  which  one  fac- 
tor is  46t,  the  other  factor  may  be  found  by  dividing 
185  tens  by  46  tens,  the  quotient,  4,  is  supposed  to  be 
the  u  of  the  root.  The  product  of  46t  by  4  =  184t, 
which  tat  en  from  the  remaining  part  of  the  power 
leave  16.  This  remainder  must  contain  the  square  of 
the  u  of  the  root.  The  sq.  of  4  =  16,  which  taken  from 
the  remaining  part  of  the  power  leave  nothing. 

54756  is  thus  found  to  be  a  square  number  the 
square  root  of  which  is  234, 

Remark.  It  is  observed  that  the  tens  and  unite  of  the  root 
as  found  are  supposed  to  be  the  correct  numbers  for  those  or- 
ders, respectively.  Sometimes  the  obtained  quotient  may  be 
so  great  that  its  product  by  the  divisor  will,  when  subtracted 
from  the  remaining  part  of  the  power,  leave  a  remainder  too 
small  to  contain  the  partial  products  that  are  yet  to  be  taken 
out.  If  a  quotient  is  too  great  a  less  quotient  must  be  used. 

Exercises. 

Evolve  the  second  root  of  4096;  9216;  7569;  .0676; 
54.76;  5476;  10201;  7744;  10509;  6889;  11025;  11236; 
3844;  .9409;  53.1441;  21025;  173056;  998001;  67305616. 


166 

242.  Table  for  Determining  the  Number  of  Terms  in 
the  Cube  Root  of  a  Given  Cube  Number. 

18=1  From  the  table  it   is  seen  that  a 

93=729  cube  number  consisting  of  three  terms 

103— 1000  or  less,  contains  one  term  in  its  cube 

993— 970299        root;  that  a  cube  number  consisting  of 

1003— 1000000      four,  five  or  six  terms   contains  two 

9993=997002999  terms   in  its  cube  root;  that  a  cube 

number  consisting  of  seven,  eight,  or 

nine  terms  contains  three  terms  in  its  cube  root ;  etc. 

243.  Principle.     There  are  as  many  terms  in  the 
cube  root  of  a  cube   number   as  there  are  periods  of 
three  terms  each   in  the   number,  counting  from  the 
interval  between  units'  and  tenths'  orders. 

liemarks.    1.  The  highest  period  in  a  cube  integer  may  con- 
tain but  one  or  two  terms. 

2.  The  terms  of  the  highest  period  of  a  cube  decimal  fraction 
may  be  wholly  or  partly  represented  by  zeros. 

3.  The  lowest  period  of  a  cube  integer  may  be  wholly  or  partly 
represented  by  zeros.    Every  period  in  a  cube  number  may  be 
partly  represented  by  zeros. 

244.  Table  Indicating  the  Lowest   Place  in  which  a 
Significant  Figure   can  Occur  in  the  Written  Cube  of 
Numbers  in  the  Orders  Named. 

u3  — -  u.  .t3  =  .th.  [For  other  indicated  pro- 

t3  =  th.          .h3  =  .m.        ducts  see  corresponding  ta- 
li8 ==  m.         .th3  =  .b.         ble,  page  160.] 
th3  =  b.    Etc.,  etc. 

245.  A  Number  Consisting  of  Tens  and  Units  Involved 
to  the  Third  Power. 

Example.    343  =  what  ? 

In  the  light  of  the  definition  of  a  3d  power,  a 
number  is  cubed  by  multiplying  its  square  by  the 
number  itself. 


167 

(30  -f  4)2      =      302  -f  2  times  30  X  4  +  42  = 

30  +  4 

302  X  4  +  2  times  30  X  4"  -f  43 

30s  +  2  times  302  X  4  +  30  X  42 

303  +  3  times  30*  X  4  +  3  times  30  X  42  +  43  =  39304 
Hence. — The  cube  of  a  number  consisting  of  tens  and  units  is 
composed  of  the  cube  of  the  tens,  plus  3  times  the  square  of  the 
tens  multiplied  by  the  units  plus  3  times  the  tens  multiplied  by 
the  square  of  the  units,  plm  the  cube  of  the  units. 

246.    A  Number  Consisting  of  Hundreds,  Tens  and 
Units,  Involved  to  the  Third  Power. 

Example.     234*  —  what  ? 
234  =  200  +  30  +  4. 

(200  +  30  +  4)2  =,  2002  +   2  times  200  X  30  -f 
302  +  2  times  (200  +  30)  X  4  +  42  [See  Art.  236.] 

Re-writing  this  formula  after  removing  the  parenthesis  we 
have — 
2002-f2  times  200x30-f-302-|-2  times  200x4-}-2  times  30X4-J-42 

Multiply  this  formula  by .  200+30+4 

and  write  the  partial  products  below. 

Upon  comparing  these  2  times  30  X  42 

partial  products  we  find  that  2  times  200  X  42 

we  have  200s  +  3  times  2002  302  X  4 

X  30  +  3  times  200  X  302  2  times  200  X  ^0  X  4 

+  303  +   3  times  2002  X    4  2002  X  4 

+  6  times  200  X  30  X  4  -f  30  X  42 

3  times  302  X4+3  times  200  2  times  302  X  4 

X42+  3  times  30  X  42+43.  2  times  200  X  30  X  4 

Factoring  the  5th,   6th  303 

and  7th  terms  of  this  formu-  2  times  200  X  302 

la   reduces  them   to  3  times  2002  X  30 

(2002  -f   2   times  200  X   30  200  X    41 

+  302)  X4.    Observing  that  2  times  200  X  30  X    4 

the   parenthetical   quantity  2  times  2002  X    4 

equals  (200  -f  30)2,  the  re-  200  X  302 

duced  expression  becomes  3  2  times  2002  X  30 

times  (200  +  30)*  X  4.  2003 


168 

Factoring  the  8th  and  9th  terms  of  the  formula 
above,  reduces  them  to  3  times  (200  -f  30)  X  42. 

Re-writing  the  formula,  substituting  3  times  (200 
-f30)3  X  4  for  the  5th,  6th  and  7th  terms,  and  3  times 
(200  -1-  30)  X  4y  for  the  8th  and  9th  terms,  we  have  as 
the  completed  formula—  2003  -f  3  times  200'  X  30  -f 
3  times  200  X  302  -f  303  +  3  times  (200  +  30)2  X  4  + 
3  times  (200  +  30)  X  42  +  43. 

Any  number  consisting  of  h,  t  and  u  may  be  cubed 
in  the  same  manner  as  the  above ;  hence,  the  cube  of 
a  number  consisting  of  hundreds,  tens  and  units  is 
composed  of  the  cube  of  the  hundreds,  plus  3  times 
the  square  of  the  hundreds  multiplied  by  the  tens, 
plus  3  times  the  hundreds  multiplied  by  the  square  of 
the  tens,  plus  the  cube  of  the  tens,  plus  3  times  the 
square  of  the  sum  of  the  hundreds  and  tens  multiplied 
by  the  units,  plus  3  times  the  sum  of  the  hundreds  and 
tens  multiplied  by  the  square  of  the  units,  plus  the 
cube  of  the  units. 

This  formula  is  abbreviated  thus  :  h3  -j-  3h2  X  t 
-f  3  h  X  t?  -f  t3  +  3  [h  +  t]2  X  u  +  3  [h  -f  t]  X  u2 
+  u3. 

The  cube  of  a  number  consisting  of  th,  h,  t  and  u 
is  composed  of  th3  -f  3  th2  X  h ' .+  3  th  X  h*  -f  h3  -f 
3(th  -f  h)2  X  t  -f  3(th  +  h)  X  t2  -f  t3  +  3(th  +  h 
+t)'  X  u  -f  3(th  +  h  +  t)  X  u2  -f  u3. 

Remarks.  1.  A  careful  study  of  the  above  forms  will  discover 
the  law  by  which  a  formula  may  be  constructed  for  the  cube  of 
a  number  consisting  of  any  number  of  orders  in  the  decimal 
scale. 

2.  The  cube  of   a  fraction  is  found  by  cubing  both  its  terms. 


169 

Evolution  of.  the  Third  Root. 
Example.     ^103823  =  what  ? 
Written  form.  Thought  form. 

103823(47  Since  there  are  six  terms  ia  the 

64  power,  there  are  two  terms  in  its  cube 

48h)39823  root ;  viz. — t   and  u. 

336  The  cube  of  a  number  consisting 

6223  of  tens  and  units  is  composed  of  \?-{- 

588  3t2X  u  +  3t  X  u2  -f-  u3. 

The  cube  of  tens  is  thousands, 
hence  the  th  of  the  power  contain 
the  cube  of  the  t  of  the  root.  The  greatest  cube  num- 
ber of  th  in  103  th  is  64  th,  the  cube  root  of  which 
is  4t.  The  power  diminished  by  64th  =  39823,  which 
contains  a  product  of  which  one  factor  is  3  times  the 
square  of  the  t  of  the  root  and  the  other  is  the  u  of 
the  root.  3  times  the  square  of  4  t  =  48h.  Since  the 
product  of  h  by  u  —  h,  the  hundreds  of  the  remain- 
ing part  of  the  power  contain  the  required  product. 
398h  -j-  48h  =  7,  which  is  supposed  to  be  the  u  of 
the  root.  7  times  48h  =  336  h,  which  taken  trom  the 
remaining  part  of  the  power  leave  6223.  This  re- 
mainder contains  the  product  of  3  times  the  tens  of  the 
root  by  the  square  of  the  u  of  the  root.  12  t  X  49 
=  588t,  which  taken  from  the  remaining  part  of  the 
power  leave  343.  This  remainder  must  contain  the 
cube  of  the  u  of  the  root.  The  cube  of  7^=343, 
which  taken  from  the  remaining  part  of  the  power 
leaves  nothing.  We  thus  find  that  103823  is  a  cube 
number  and  that  47  is  its  cube  root. 

Remarks.  1.  In  the  light  of  the  foregoing  form  the  obser- 
vant pupil  will  readily  evolve  the  third  root  of  any  cube  num- 
ber. 

2.  For  the  application  of  cube  root  in  determining  lines  see 
text  books  on  Arithmetic. 


170 

Exercises. 

Evolve  the  third  root  of— 32768 ;  68921 ;  148877  ; 
250047;  287-496;  328.509;  .389017;  .438976  ;  1061208; 
1092727;  1124864;  1157.625;  1.191016;  870983875; 
12977875  ;  5735339 ;  300763 ;  912673. 


Cube  Boot  by  Endings. 

Example.  12167  is  a  cube  number;  what  is  its 
cube  root  ? 

Solution,  a.  Since  7  is  the  units  of  the  cube,  3 
must  be  the  units  of  its  cube  root,  • 

b.  The  greatest  cube  in  12,  (the  left  period)  is  8,  the 
cube  root  of  which  is  2 ;  hence  2  is  the  tens  of  the  cube 
root.  The  required  root  is,  therefore,  23. 


SECTION  XV. 
TEST  PROBLEMS. 

1.  If  a  merchant  mark  his  goods  25%  above 
cost,  and  sell  them   at  25  %   below  the  marked  price, 
does  he  gain  or  lose  and  at  what  rate  %  ? 

2.  A  load  of  hay  weighs  15  cwt.21  lb. ;  if  2  cwt. 
11  lb.  be  sold,  what  part  of  the  load  remains  ? 

3.  Multiply  .025  by  2.5. 

4.  A  owned  f  of  a  store  and  sold  to  B  -J  of  his 
share  and  to  C  f  of  his  share;   what  part  of  it  did  he 
still  own  ? 

5.  Eequired  the  cost  of  I  of  a  yard  of  cloth,  if  | 
of  a  yard  cost  $7f . 

6.  The  longitude  of  Washington  is  76°  56'  west 
of  London,  what  change  would  it  be  necessary  to  make 
in  a  time-piece  in  coming  from  London  to  Washington? 


171 

7.  At  what  rate  per  cent,   will  $380  in  7  yr.  3 
mo.  yield  $165.30  interest  ? 

8.  £  =  what  part  of  9  ?      What  per  cent,  of  9  ? 

9.  If  9 £  eggs  weigh  a  pound,  and  a  pound  of  eggs 
equal  a  pound  of  steak  as  food,  at  what  price  per  dozen 
must  eggs  be  bought  in  place  of  steak  at  22  /  per  Ib.  ? 

10.  What  is  the  weight  of  the  air  in  a  room  25  ft. 
by  20  ft.  by  12  ft.,  water  weighing  770  times  as  much 
as  air,  and  a  cubic  foot  of  ^vater  weighing  1000  oz. 
Avoirdupois  ? 

11.  If  lead  is  11.445  times  as   heavy  as  water, 
what  is  the  weight    of  a  piece  of  lead  1m.   by  2  dm. 
by  5  cm.  ? 

12.  A  lot  of  goods  was  marked  40%  above  cost; 
if  sold  at  30%  less  than  the  marked  price,  was  there  a 
gain  or  loss   and  at  w  hat  rate  per  cent.  ? 

13.  Which  would  yield  the  better  pay,  a  1%  bond 
at  115  or  a  6%  bond  at  98,  and  at  what  rate  better? 

14.  If  a  merchant  sell  flour  at  $9  per  barrel,  and 
wait  6  months  for  his  pay,   at  what  price  could  he  af- 
ford to  sell  for  cash  if  money  is   worth  2%   a  month? 

15.  For  what  sum  must  a  3  months'   note  be 
drawn  so  that  when  discounted  by  a  bank  at  7  per  cent. 
I  may  get  $400  ? 

16.  At  what  quotation  must  I  buy  a  6%  stock  to 
make  as  good  an  investment  as  from  a  4%  stock  at  75  ? 

17.  A  traveling  salesman  is  allowed  12  %    com- 
mission on  his  sales  ;   his   employer's  rate   of  profit   is 
20%  on  the  goods  sold;  what  is  the  first  cost  of  goods 
which  the  salesman  sells  for  $7.66  ? 

18.  A  water  tank   is  3  ft.  deep,  4  ft.  long,  and  4 
ft.  wide ;  it  is  supplied  from  a  flat  roof  20  ft.  by  30  ft.  ; 
what  depth  of  rain  must  fall  to  fill  the  tank  ? 

19.  Eequired  the  present  worth  of  $1320  due  in 
3  yr.  4  mo.  without  interest,  if  money  is  worth  6  % . 


172 

20.  A  man  bought  a  horse  for  $72  and  sold  it  for 
25  per  cent,  more  than  it  cost  and  10%  less  than  the 
asking  price ;  what  did  he  ask  for  the  horse? 

21.  2  ft.  9  in.  =  what  part  of  a  rod  ? 

22.  Cincinnati  is  7°  49'  west  of  Baltimore;  when 
it  is  noon  at  Baltimore,  what  time  is  it  at  Cincinnati  ? 

23.  A  man  bought  stock  at  25%  below  par,  and 
and  sold  it  ai  25%  above  par;  required  his  rate  of  gain. 

24.  How  many  yards  of  carpet  f  of  a  yard  wide 
will  cover  a  floor  18  ft.  by  15  ft.  ? 

25.  I  -5-  f  =  what? 

26.  Reduce  to  equivalent  fractions  with  1.  c.  d. 
§,  f  and  f . 

27.  How  find  the  area  of  a  circle  if  only  the  ra- 
dius be  given? 

28.  If  a  wheel  turn   17°  30'  in  35  minutes,  in 
what  time  will  it  make  a  revolution  ? 

29.  If  rosin  be  melted  with  20%  of  its  weight  of 
tallow,  what  %  of  the  weight  of  the  mixture  is  tallow? 

30.  Eequired  the  weight   in   kilograms  of  a  bar 
of  iron  3.6  m.  long,  6  cm.  wide,  and  2  cm.  thick,  if  iron 
is  7.8  times  as  heavy  as  water. 

,  31.  With  gold  at  103  what  rate  of  interest  do  I 
make  on  a  $1000  5-20  bond  bought  at  106? 

32.  A  and  B  are  partners  for  1  yr.,  A  putting  in 
$2000  and  B  $800 ;  how  much  more  must  B  put  in  at 
the  end  of  6  months  to  receive  one-half  the  profits  ? 

33.  How  many  grams  does  a  Dl.  of  water  weigh? 

34.  Demonstrate  the  principle: — The  product  of 
the  means  of  a  proportion  equals  the  product  of  the 
extremes. 

35.  The  surface  of   the  earth  contains   about 
144000000  sq.  mi.  of  water,  and  about  53000000  sq,  mi. 
of  land .    What  %  of  the  earth's  surface  is  water  ? 


173 

36.  f  X  f  =  what  ? 

37.  If  a  watch  sell  for  $60  at  a  loss  of  22%,  for 
what  should  it  sell  to  gain  30  per  cent.  ? 

38.  A  broker  bought  stock  at  8%   premium  and 
sold  it  at  9%  discount  thereby  losing  $510;  how  many 
shares  did  he  buy  ? 

39.  What  is  the  net  tax  in  a  town  whose  taxa- 
able  property  is  assessed  at  $430000,  at  12   mills  per 
dollar,  5  per  cent,  being  paid  for  collection  ? 

40.  The  difference  of  time  between  two  places   is 
2  hr.    15   min.    10   sec. ;    required   their  difference  in 
longitude. 

41.  Eeduce  .037  lb.  Av.  to  drams. 

42.  Reduce  84.5  ars.  to  square  meters. 

43.  An  agent  received  $484.50  with  which  to  buy 
sheep   after  deducting  his  commission  at  2%  ;  how 
much  money  did  he  spend  for  sheep  ? 

44.  A  policy  for  $2675  cost  $53.30 ;  find  the  rate 
of  insurance. 

45.  The  difference  of  time  between  two  places  is 
45  min.  30  sec.,  and  the  place  having  the  earlier  time 
is  in  longitude  85°  40'  west;  required  the  longitude  of 
the  other  place. 

46.  Reduce  .096  of  a  bu.  to  the  decimal  of  a  pint. 

47.  A  note  of  $125  dated  May  3, 1883,  and  payable 
in  sixty  days,  with   interest   at  5%,   was  discounted 
June  18,  1883,  at  10%  ;  required  the  proceeds. 

48.  How  many  bu.  of  wheat  will  fill  a  bin  8  ft. 
by  5  ft.  by  4  ft.  ? 

49.  How  many  meters  of  carpet  .7  m.  wide  will 
cover  a  floor  4  m.  by  4.5  m.  ? 

50.  Add  |  f  and  f 

51.  Multiply  f  by  2.3. 

52.  Divide  2.3  by  f . 


174 

53.  f  =  what  part  of  2.3?    What  %  of  2.3? 

54.  Reduce  135  sq.  rd.  54  sq.  ft.  to  the  decimal  of 
an  acre. 

55.  How  much  money  must  I  remit  to  my  agent 
to  buy  goods  and  pay  himself  $24  commission  at  1£%? 

56.  A  sold  a  horse  to  B  and  gained  \  of  its  cost ; 
B  sold  it  for  $80  and  lost  \   of  what   it  cost  him ;  how 
much  did  A  pay  for  the  horse  ? 

57.  Divide  3.45  by  1.5. 

58.  A  garden  contains  800  sq.  rd.   and  is  33^  rd. 
long;  how  wide  is  it  ? 

59.  What  per  cent,  of  f  is  f  ? 

60.  What  is  J  per  cent,  of  .5  ? 

61.  428  is  1%  more  than  what  number? 

62.  A  man  owes  $300  due  in  4  mo.,  $600  due  in  5 
mo.,  and  $100  due  in  6  mo. ;  if  he  pay  i  of  his  indebt- 
edness in  2  mo.,  when   in  equity  should  he   pay   the 
balance  ? 

63.  What  is  the  difference  between  the  true  and 
the  bank  discount  of  $359.50  for  90 days  without  grace, 
money  being  worth  8%  ? 

64.  A  board  is  20  ft.  long  and  9  in.  wide  ;  what  is 
it  worth  at  $30  per  M  ? 

65.  At  what  time  between    6  and  7  o'clock  are 
the  hour  and  minute  hands  of  a  watch  together? 

66.  A  broker  bought  60  shares  of  stock  at  106J, 
received  a  5%  dividend  and  then  sold  at  104;   did  he 
gain  or  lose  and  how  much  ? 

67.  How  many  ft,,  board  measure,  each  board  be- 
ing 18  inches  wide,  can  be  cut  from  a  squared  log  16 
ft.  long,  18  in.  wide  and  10  in.  thick,  allowing  £  in.  for 
each  cut  of  the  saw  ? 

68.  If  12  men  can  do  a  piece  of  work  in  5£  days, 
in  how  many  days  can  8  men  and  5  boys  do  it,  1  man 
doing  the  work  of  2?  boys  ? 


175 

69.  If  a  note  of  $5000,  for  4  mo.,  at  6%  int.  per 
annum,  be  discounted  in  bank,  on  day  of  making,  at 
at  8%  per  annum,  what  will  be  the  proceeds? 

70.  If  it  cost  $312  to  fence  a  field  216   rd.  by  24 
rods,  what  cost  the  fence   of  a   sq.  field  of  equal  area? 

71.  A  merchant  bought  goods  at  20  cents  per  yd, 
and  sold  them  at  40%    profit,  after  allowing  his  cus- 
tomers 12£%  discount  off;  what  was  the  marked  price? 

72.  A  vessel,  at   noon,  sails  due  north ;  after  a 
certain  time  an  observation  shows  the  sun  to  have 
sunk  toward  the  west  2  signs  15  degrees ;  how  long  has 
the  vessel  been  sailing  ? 

73.  I  sell  a  bill  on  London  for   £1675,  at  the  rate 
of  24.3  cents  per  shilling  ;  how  much  do  I  receive? 

74.  If  & 6000  of  6%   stock  be  sold  at  90,  and  the 
proceeds  invested  in  10%  stock  at   105,   what  will  be 
the  change  in  the  income? 

75.  I  buy  $1500  worth  of  goods  at  4  mo.,  $850  at 
3  mo.,  $1750  at  5  mo. ;  what  is  the  equated  time  for  the 
payment  of  the  whole  ? 

76.  The  square  root  of  .1369  plus  the  square  root 
of  1296  equals  what? 

77.  A  owesB  $1800;  B  offers    to  allow  5%  off 
for  cash ;    A  pays  $1425,  how  much  is  still  due  ? 

78.  Evolve  the  second  root  of  15625. 

79.  What  is  the  surface  of  a  cube  which  contains 

8  times  the  volume  of  a  cube  whose  edge  is  -f  of  a  foot  ? 

80.  What  is  the  capacity  of  a  cylinder  20ft.  long, 
whose  radius  is  2  ft.  ? 

81.  I  bought  goods  in  Europe,  paid  20%  duties, 
a  commission  of  2%   upon  duties  and  cost,   and  sold 
them  at  $10  per  yard,  clearing  34%   on  invoice  price; 
required  the  entire  cost  per  yard. 

82.  A  travels  5^  hours  at  the  rate  of  6  miles  per 
hr.,  B  then  follows  from   the  same  point  at  the  rate  of 

9  mi.  per  hr. ;  how  long  will  it  take  B  to  overtake  A  ? 


176 

83.  At  24.2  cents  per  shilling,  what  cost  £1050? 

84.  What  principal  in  3  yr.  4  mo.  24  da.  at  5% 
will  amount  to  $761. 44? 

85.  For  what  sum  must  a  note  dated  April  5, 1883, 
for  90  da.   be   drawn,   that  when  discounted   at  7%, 
April  21,  1883,  the  proceeds  may  be  $650? 

86.  A  room  is  26  ft.  long,  16  ft.  wide  and  12  ft. 
high  ;  what  is  the  distance  from  one  of  the  lower  cor- 
ners diagonally  to  the  opposite  upper  corner  ? 

87.  St.  Petersburg  is  30°  19'  east  longitude,  and 
Indianapolis  is  86°  5'  west  longitude;  when  it  is  3  a. 
m.  at  St.  Petersburg,  what  is  Indianapolis  time  ? 

88.  Reduce  492  dekagrams  to  quintals. 

89.  How  many  bricks,  each  8  in,  long,  4  in.  wide 
and  2?  in  thick,  will  be  required  for  a  wall  120  ft.  long 
8  ft.  high  and  1  ft.  4  in.  thick,  no  allowance  being  made 
for  mortar  ? 

90.  If  6  men  cai.  build  a  wall  20  ft.  long,  6  ft.  high 
and  4  ft.  thick  in  16  days,  in  what  time  can  24  men 
build  a  wall  200  ft.  long,  8  ft.  high  and  4  ft.  thick  ? 

91.  A  man  bought  a  square  farm  containing  140 
acres  and  100  sq.  rd. ;  required  the  length  of  one  side. 

92.  Evolve  the  3rd  root  of  592704.    Of  2985984. 

93.  A  farmer  exchanged  100  bu.  of  wheat  at  $1.25 
per  bu.  for  corn  at  $.37-J  per  bu.;  how  many  bu.  of  corn 
did  he  receive  ? 

94.  The  sum  of  two  numbers  is  785 ;  their  differ- 
ence is  27 ;  what  are  the  numbers  ? 

95.  Divide  Yr8  b7  iV 

96.  Reduce  •£$  to  a  decimal  fraction. 

97.  How  many  loads  are  contained  in  a  pile  of 
wood  40.16  ft.  long,  7.04  ft.  high  and  4  ft.  wide,  if  each 
load  contains  1 J  cords  ? 

98.  If  I  exchange  $12000  of  8%   stock  at  115  for 


177 

5%  stock  at  69,  do  I  gain  or  lose  on  annual  income, 
and  how  much  ? 

99.  How  large  a  sight  draft  can  be  bought  for 
$259.52,  exchange  being  lf%  premium  ? 

100.  What  are  the  cubical   contents   of  a  cylinder 
that  will  just  enclose  a  sphere  9  in.  in  diameter? 

101.  Evolve  the  second  root  of  -^  to  two  decimal 
places. 

102.  Three  farms  contain  respectively,  356,  898, 
and  1254  acres,   which  I  desire  to  cut  into  building 
lots  of  the  largest  equal  size  possible  ;  how  many  acres 
will  each  lot  contain  ? 

103.  Divide  f  by  f . 

104.  Multiply  .803  by  .03. 

105.  The  sun  at  12  o  clock  is  over  the  meridian  at 
Washington;  over  what  meridian  will  it  be  after  trav- 
eling through  5  signs  and  5  degrees  ?     What  time  will 
it  then  be  at  Washington  ? 

106.  How  many  grams  does  a  liter  of  rain  water 
weigh  ? 

107.  At  7  cents  per  square  foot,  required  the  cost 
of  a  brick  walk  6  feet  wide  around  a  lot  200  ft.  by  300  ft. 

108.  What  sum  of  money  loaned  ut  6%  for  10  mo. 
will  yield  as  much  interest  as  $750  loaned  at  4%  for 
11  months  ? 

109;  Required  the  area  of  a  circle  whose  radius  is 
10  feet. 

110.  A,  B  and  C  eat  8  loaves  of  bread  of  which  A 
furnishes  3,  and  B  5.    C  pays  A  and  B8  pieces  of  mon- 
ey of  equal  value  ;  how  should  they  divicle  the  money? 

111.  For  what  sum  must  I  make  a  bank  note  for 
60  days,  which  discounted  at  10%    will  pay  a  $1000 
debt  now  due  ? 

112.  A  street  60  ft.  wide  is  crossed  at  right  angles 
by  another  80  ft.  wide,  what  is  the  distance  between 
diagonal  corners? 


178 

113.  A  wall  is  83  meters  long,  2  Dm.  high,  and  5 
dm.  thick ;  what  is  its  value  at  $3.30  per  cu.  meter?- 

114.  a.     What  is  the  value  of  the  N.  E.  \  of  the  N. 
W.  }  of  section  16,  at  $12  per  acre  ? 

6,     Make  a  plat  of  the  congressional  township 
and  indicate  the  part  described. 

115.  A  miller  takes  for  toll  4  quarts  from  every  5 
bu.  of  grain  ;  what  per  cent,  does  he  get  ? 

116.  In  what  time  will  $375.40  yield  $37  54  inter- 
est  at  6%  per  annum? 

117.  A  barri  is  40  ft.  wide;  the  comb  is  15  ft.  from 
the  plate  and  the  rafters  are  of  equal  length  ;  what  is 
the  length  of  each  rafter  ? 

118.  If  money  is  worth  12%,  what  is  the  true  dis- 
count of  $235. ]0,  due  one  year  hence  ? 

119.  In  a  cube  whose  edge  is  £  in.,  how  many  cubes 
each  ^  of  an  inch  wide  ? 

120.  A  rectangular  field  15   rods  wide  contains  3 
acres ;  how  long  is  it  ? 

121.  How  many  yards  of  carpet  27  inches  wide 
will  cover  a  floor  18  ft.  long  by  14  ft.  wide  ? 

122.  If  an  article  be  sold  for  twice  its  cost,  what 
is  the  rate  per  cent,  of  gain  ? 

123.  If  an  article  be  sold  for  one-half  its  cost,  what 
is  the  rate  per  cent,  of  loss  ? 

124.  If  an  article  is  sold  for  one  third  its  cost,  what 
is  the  rate  per  cent,  of  loss? 

125.  Property  worth  $8760  is  rented  for  $650  per 
annum;  what  rate  of  interest  does  the  investment 
yield? 

126.  In  what  time  will  $2450  yield  $725  int.  at  S%? 

127.  A  man  recei"es  $280  interest,  annually,  on  a 
1%  loan;  what  is  the  face  of  his  loan? 

128.  What  principal  will  amount  to  $5750  in  3  yr. 
5  mo.  17  da.  at  Q%  ? 


179 

129.  For  what  ^sum  must   property  worth  $6000 
be  insured  at  5%  to*  cover  f  of  the  property  and  the 
premium  ? 

130.  What  is  the  selling  price  of  corn  whoso  first 
cost  is40cts.  per  bu.,  freight  8%  and  rate  of  gain  16|-%? 

131.  A   merchant  sold  two  bills  ot  goods  for  $50 
each;  on  one  he  gained  15%,  and  on  the  other  he  lost 
15%  ;  required  his  gain  or  loss. 

132.  What  is  the  rate  of  interest  on  an  investment 
in  U.  S.  4  per  cents,  at  95  ? 

133.  A  merchant  is  offered  goods  for  $2500  cash,  or 
for  $2650  on  60  days  time  ;  which  is  the  better  offer  if 
money  is  worth  8%  per  annum? 

134.  For  what  must  5%   stock  be  bought  that  it 
may  yield  7%  interest  on  the  investment  ? 

135.  U.  S.  3f  s  bought  at  98  pay  what  rate  of  in- 
terest on  the  investment  ? 

136.  What  sum  of  money  will  yield  as  much  in- 
terest in  10  months  at  6%,  as  $1500  will  yield  in  12 
mo.  at  4%  ? 

137.  How  much  must  be  paid  in  U.  S.  currency 
for  a  draft  of  £210  10s.   6d.,  exchange  being  102,  bro- 
kerage £%,  and  gold  at  104? 

138.  Reduce  -f,  f,  -^  to  equivalent  fractions  having 
the  1.  c.  d. 

139.  Multiply  §  by  f ;  -fT  by  .04;  2.06  by  ;0013. 

140.  Divide  -?-  by  f ;  .004  by  f ;  3.064  by  .08. 

141.  Reduce  5  yd.  2  ft.  7  in.  to  inches. 

142.  Reduce  I  mile  to  lower  integers. 

143.  A  box  is  4|  m,  long,  8  dm.  wide  and  3.5  dm. 
deep ;  how  many  Kg.  of  distilled  water  will  it  hold  ? 

144.  A  pile  of  wood  is  15.5  m.  long,  12  dm.  wide, 
and  1.8  meters  high;  how  many  sters  does  it  contain? 

145.  f  ==what  %  off? 


180 

146.  Reduce  -£$  to  the  decimal^ scale. 

147.  Two  men  paid  $150  for  a  horse  ;  one  paid  $90 
and  the  other  paid  $60 ;  they  sold  it  so  as  to  gain  $75 ; 
what  was  the  share  of  each  ? 

148.  A  man  owes  A  $105,  B  $75.  and  C  $120 ;  he 
has  only  $125  ?  how  much  should  he  pay  to  each? 

149.  Find  g.  c.  d.  of  348  and  1116. 

150.  What  is  the  face  of  a  note  payable  in  90  days, 
on  which  $3500  can  be  obtained  at  a  bank,  discounting 
at  6jg? 

151.  An  irregular  mass  of  metal  immersed  in  a 
vessel  full  of  water  caused  2.25  1.  of  water  to  overflow 
the  sides  of  the  vessel ;  what  was  the  weight  of  the 
metal  if  its  specific  gravity  was  7.2  ? 

152.  What  is  the  price,  at  $11.65  per  kilogram,  of 
1  liter  of  alcohol,  it£  s.  g.  boing  .79? 

153.  What  is   the  capacity  in  liters,  of  a  cylin- 
drical cup  16  cm.  in  diameter  and  1.3  dm.  deep? 

154.  A&  B  gain  in  business  $4160,   of  which  A  is 
to  have  8%  more  than  B,  how  much  will  each  receive? 

155.  A  merchant  insures  a  cargo  ol  goods  for  $3456 
at  3£%'  the  policy  covering  both  property  and  premi- 
um ;  what  was  the  value  of  the  property  ? 

156.  If  a  stack  of  hay   82   ft.  high,  weigh  7  cwt., 
what  weighs  a  similar  stack  15  ft.  high  ? 

157.  For  what  sum  must  a  note  be  drawn  at  3  mo. 
that  the  proceeds,  when  discounted,  without  grace,  at 
a  bank,  at  8%  shall  be  $1274? 

158.  The  contents  of  a  cubical  block  of  stone  are 
4913  cu%  cm. ;  required  its  superficial  contents  in  sq.  m. 

159.  I  imported  12  casks  of  wine,  each  containing 
48  gallons  invoiced  at  $2.75  per  gal. ;  paid   $108  for 
freight,  and  an  Ad  Valorem  duty  of  36%  ;  what  is  my 
rate  of  gain  if  I  sell  the  whole  for  $3258  ? 


181 

160.  A  New  York    merchant  gave  $960  for  a  bill 
of  £250  on  London,  required  the  rate  of  exchange. 

161.  What  is  the  largest  square   stick   that  can  be 
cut  from  a  log  4  ft.  in  diameter  ? 

162  A  sea  captain  set  his  watch  at  London,  and 
found  after  traveling  that  it  was  4  hours  faster  than 
the  local  time  of  the  place  he  had  reached  ;  how  far  had 
he  traveled  and  in  what  direction  ? 

163.  What  is  the  area  of  a  field    whose   parallel 
sides  are  90  rods  and  124  rods  long,  respectively,  and 
the  perpendicular  distance  between  them  is  50  rods  ? 

164.  A  has  $550,  and  B  has   $330 ;  what  per  cent, 
of  the  money  of  each  is  the  money  of  the  other  ? 

165.  A  farmer's  wagon  loaded  with  wheat  weighed 
4912  lb.,  and  his   wagon   alone   weighed   920  lb.,  what 
was  his  load  of  wheat  worth  at  95  cts.  per  bu.  ? 

166.  If  a  bolt  of  paper  is  8  yd.  long  and  J  yd.  wide, 
how  much  will  the  paper  cost  at  30  cts.  per  bolt,  to  pa- 
per the  walls  and  ceiling  of  a   room   18  It.   long,  15  ft. 
wide  and  10  ft.  high  ? 

167.  A  man  paid  $30.09   for  the   use   $204   for  11 
mo.  ?  what  was  the  rate  of  interest  ? 

168.  What  is  now  due  on  a  note  given  Sept.  ], 
1881,  the  principal  being  $430,  the  rate 7%,  per  annum, 
endorsed  Aug,  4,  1882,  $34,  July  7,  1883,  $118? 

169.  A  railroai    right  of  way    4  rods  wide  passes 
diagonally  through  a  quarter  section  of  land ;  ho  w  many 
acres  remain  unoccupied  by  the  road  ? 

170.  Reduce  f  da.,  3  min.  to  the  decimal  of  a  day. 

171.  A  man  owned  a  farm  of  40  A.  3  R.  22  Sq.  rd., 
and  sold  it  at  $45.50  per  acre ;  what  sum  did  he  realize? 

172.  The  selling  price  of  a  house  was  130%  of  the 
cost ;  the  gain  was  $240;  required  the  cost. 

173.  The  interest  was  $45,  the  rate  8%  per  annum, 
the  time  1  yr.  3  mo.  18  da.  ;  required  the  principal. 


182 


174.  .Reduce  6  oz.,  Av.  to  the  decimal  of  a  cwt. 

175.  How  many  boards  16  ft.  long  will  be  required 
to  fence  a  lot  125  ft.  by  80  ft.,  the  fence  being  5  boards 
high  ?     .Required  its  cost  at  $27  per  M,  each  board  be- 
ing 5  in.  wide  and  1  in.  thick. 

176.  Divide  sixteen  ten-miilionths  by  twenty-five 
ten-thousandths. 

177.  How  many  bricks  can   be  laid   in  a  pavement 
25  ft.  4  in.  long  by  15  ft.  3  in.  wide,  each  brick  being  8 
in.  by  4  in.  ? 

178.  When  it  is  2  o'clock  at  Terre  Haute,  what  is 
the  time  47°  13'  27"  west  of  Terre  Haute  ? 

179.  The  capacity  of  a  cubical  cistern  is  74088  cu. 
in. ;  how  many  sq.  ft.  in  the  bottom  of  it  ? 

180.  What  cost  2  Ib.  4  oz.  of  cloves   at  87£  cents 
per  Ib.  ? 

181.  At  $11  per  rod,  what  will  be  the  cost  of  fenc- 
ing a  lot  in  the  form  of  a   right  angled  triangle,  whose 
sides  forming  the  right  angle  are  264  rods  and  23  rods, 
respectively  ? 

182.  A  grocer  bought  a  hhd.  of  sugar  weighing  3 
cwt.  2  qr.  20  Ib.  for  $30,  and  sold  it  at  7J  Ib.  for  $1  ; 
what  per  cent,  does  he  gain  ? 

183.  A  6  months'  note  for  $5000,  drawing  interest 
at  6%  per  annum,  was  discounted — not  in  bank — 3  mo. 
after  date  at  10%  per  annum  ;  required  the  proceeds. 

184.  A  physician  bought  2  Ib.  9  oz.  of  quinine  at 
$38  per  Ib.,  Avoirdupois,  and  sold  it  in   10  grain  doses 
at  20  cts  per  dose  ;  did  he  gain  or  lose  and  how  much  ? 

185.  Find  the  Amt.  of  $3350  for  3  yr.  9  mo.  at  8%. 

186.  If  you  travel  360  miles  in  12   days  ot  8  hr. 
each,  how  many  miles  can  you  travel,  at  the  same  rate, 
in  60  days  of  6  hr.  each  ? 

187.  What  is  the  side  of  a  cube    which  contains  as 
many  cu.  it.  as  a  box  8  ft.  3  in.  long,  3  ft.  wide,  and  2 
ft.  7  in.  deep  ? 


183 


188.  What  per  cent,  of  14  bu.  is  5  bu,  3  pk.  5  qt.  ? 

189.  Find  the  g.  c.  d.  of  368  and  612. 

190.  I  =  what  decimal  fraction  ? 

191.  If  50  men  build  50  rods  of  wall   in  75  days 
how  many  men  will  build  80  rods  of  similar  wall  f  as 
thick  and  £  as  high  in  40  days  ? 

192.  A  farm  is  125  rods  square  and  a  rectangular 
field  containing  the  same  area   is    130   rods   long;  how 
wide  is  it  ? 

193.  By  what  must  I  be  multiplied  that  the  pro- 
duct may  be  12  ? 

194.  A  flash  of  lightning  is  seen  47  sec.  before  its 
peal  ot  thunder  is  heard ;  required  the  distance  of  the 
thunder  cloud. 

195.  What  is  the  difference  betweeen  the  simple  in- 
terest of  $1000  at  6%,  for  5  yr.,  and  the  true  discount 
of  the  same  sum  for  the  same  time  at  the  same  rate  ? 
Which  is  the  greater  and  how  much  ? 

196.  What  is  the  value  in  English  money,  of  $500 
at  $4.84  per  £  ? 

197.  How  many  square  yards  of  canvas  would  be 
required  to  make  a  balloon  of  globular  form,  20  yards 
in  diameter? 

198.  If  linen  is  bought  at  2  s.  9  d.  per  yd.,  for  what 
must  it  be  sold  to  gain  25%  ?  What  is  the  selling  price 
in  U.  S.  money  ? 

199.  If  the  fore  wheels  of  a  wagoii  be  4  feet  in  di- 
ameter and  the  hind  wheels  5  feet  in  diameter,   how 
many  more  revolutions  will  the  former  make  than  the 
latter  in  going  a  mile  ? 

200.  On  a  base  of  120  rods,  a  surveyor  wishes  to 
lay  off  a  rectangular  field  that  shall  contain  60  A.  ;  how 
far  from  his  base  line  must  he  run  out  ? 

201.  A  sells  to  B,  tea   worth    45/   for  48/;    what 
should  B  charge  A  for  sugar  worth   9/  to   balance  the 
transaction  ? 


184 

SECTION   XVI. 

REVIEW  QUESTIONS  AND  TOPICS. 
Define  mathematics. — What  is   conditional   for 
extension? — What  branches  of  knowledge  involve 
a  study  of  extension  ? — What  is  known  of  every 
conscious  mental  state  ? — How  are  mental  states 
5     known   to   be  distinct  ? — How    does   the   idea   of 
number  arise? — What  is  conditional  for  the  num- 
erical idea? — Define  Arithmetic  as  a  science;   as 
an  art.— What  attribute    furnishes    the   basis    of 
number  in   particular? — By    what    process   does 

10  the  mind  obtain  the  idea  one  ? — Define  the  integral 
unit,  or  unit  one. — How  many  classes  of  second^ 
ary  ones  are  mentioned  ? — Define  a  fractional  unit ; 
a  multiple  unit. — What  is  the  primary  idea  in  Ar- 
ithmetic?— How  are  other  units  related  to  the  unit 

15  one? — What  is  a  unit  object? — Define  a  number. 
— Of  what  units  may  a  number  be  composed  ? — On 
what  basis  are  numbers  classified  as  integers  and 
fractions? — Define  an  integer;  a  fraction. — On 
what  basis  are  numbers  classified  as  abstract  and 

20  concrete  ? — Define  an  abstract  number ;  a  con- 
crete number. — On  what  basis  are  abstract  inte- 
gers classified  as  prime  and  composite  ? — On  what 
basis  are  numbers  classified  as  simple  and  de- 
nominate?— Define  a  simple  number;  a  denomin- 

25  ate  number;  a  compound  number. — How  is  a  de- 
nominate number  sometimes  defined  ? — What  ob- 
jection ? — How  is  a  compound  number  sometimes 
defined?— What  objection  ?— Define  notation.— 
What  kinds  are  usually  presented  in  Arithmetic  ? 

30  —What  is  the  alphabet  of  the  Eoman  notation  ?— 
What  does  each  letter  signify  ? — What  are  the 
limits  of  the  Boman  notation? — State  the  princi- 


185 

pies. — What  is  the  alphabet  of  the  Arabic  nota- 
tion?— What  is   the  signification  of  each  charac- 

35  ter? — How  many  and  what  distinct  systems  of 
numbers  make  use  of  the  Arabic  characters? — 
Define  a  scale;  the  decimal  scale? — What  is  a 
unit  of  the  first  order? — How  are  units  of  dif- 
ferent orders  formed? — What  are  periods  in  the 

40  decimal  system  of  numbers? — What  orders  are 
embraced  by  any  period? — What  is  meant  by  a 
decimal  division  of  1? — How  are  lower  orders  ot 
units  formed? — State  the  principle  of  the  decimal 
scale. — What  is  the  office  of  the  decimal  point? — 

45  Describe  units'  place. — How  many  and  what  kind 
of  units  may  be  expressed  therein? — Describe  tens' 
place. — How  many  and  what  kind  of  units  may  be 
expressed  therein? — How  may  higher  decimal 
units  be  expressed? — How  may  lower  decimal  units 

50  be  expressed? — What  is  the  representative  scale? — 
What  is  the  simple  value  of  a  figure? — What  is  the 
local  value  of  a  figure? — What  are  the  limits  of  the 
decimal  system  of  numbers? — Define  numeration ; 
reading  numbers. — In  reading  a  number  where 

55  should  the  word  and  be  used? — What  elements 
constitute  the  primary  idea  of  a  fraction? — How 
many  and  what  numbers  are  necessary  to  the  idea 
of  a  fraction? — What  are  these  numbers  together 
called? — Define  the  denominator;  the  numerator. 

60  —What  is  the  denomination  of  a  fraction? — What 
exception? — How  may  decimal  fractions  be  notat- 
ed? — How  are  other  fractions  notated? — What 
names  are  applied,  respectively,  to  the  terms  of  a 
written  fraction? — How  is  a  compound  number 

65  thought? — In  compound  numbers  the  number  of 
denominations  is  how  limited? — What  kind  of  a 


186 

scale  has  each  of  the  common  measures? — Each  de- 
nominate number  forming  a  part  of  a  compound  num- 
ber is  how  thought? — How  is  a  compound  number 

70  notated? — How  are  the  parts  of  a  compound  num- 
ber written  as  to  abbreviation  and  punctuation?— 
How  is  a  compound  number  read? — Define  reduc- 
tion; descending;  ascending;  and  state  how  each 
is  effected? — Define  computation? — How  are  num- 

75  bers  compared  previous  to  computing? — What 
mental  act  effects  a  computation? — How  many  and 
what  operations  does  the  mind  perform  upon  num- 
bers?— What  is  the  primary  judgment  in  computa- 
tion?— What  is  an  equation? — What  are  the  mem- 

80  bers  of  an  equation? — Define  sum,  addition,  addends. 
— State  the  mental  acts  involved  in  addition. — State 
and  illustrate  the  principles  of  addition. — Describe 
the  sign  of  addition  and  state  its  use. — Show  by  an 
example  that  reduction  may  be  necessary  in  addi- 

85  tion. — Which  reduction  may  be  involved  in  addi- 
tion?— Define  product. — What  other  term  means 
the  same? — Define  multiplication. — State  the  gene- 
sis of  multiplication. —  What  act  of  synthesis  is 
involved? — What  is  the  act  of  multiplication?— 

90  Define  multiplicand  ;  multiplier. — If  the  multiplier 
is  an  integer  what  relation  do  multiplicand  and 
multiplier  sustain? — If  the  multiplier  is  a  fraction 
in  what  consists  the  multiplication? — Define  a  fac- 
tor.— State  and  illustrate  each  of  the  ten  principles 

95  of  multiplication. — Describe  and  state  the  use  of 
the  sign  of  multiplication. — Which  reduction  is 
often  involved  in  multiplication? — Show  by  exam- 
ple how  reduction  may  be  necessary  in  multiplica- 
tion.— What  is  meant  by  "continued  product?"- 

100  Define  a  composite  number ;  define  composition ;  a 


187 

prime  number. — Under  what  conditions  are  num- 
bers relatively  prime? — Define  a  common  multiple  ; 
the  1.  c.  m.;  a  common  measure;  the  greatest 
common  measure. — State  and  illustrate  each  of  the 

105  principles  given  under  composition. — Define  a 
power;  involution;  a  root;  second  power ;  second 
root. — How  are  higher  powers  formed  and  named? 
— How  are  roots  named? — What  is  the  first  power 
and  first  root  of  a  number? — Describe  and  define 

110  the  index  of  a  power. — Repeat  the  table  of  squares 
given ;  the  table  of  cubes. — State  and  apply  rules 
for  squaring  numbers. — Under  how  many  and  what 
heads  is  the  synthesis  of  numbers  discussed? — How 
many  and  what  methods  of  synthesis  are  there? — 

115  What  relation  do  the  other  processes  called  syn- 
thetic sustain  to  the  one  method  of  synthesis? — 
State  in  substance  the  last  four  remarks  given 
under  synthesis? — On  what  ground  is  a  number 
resolvable  into  parts? — Define  difference,  giving 

120  two  definitions. — Define  subtraction;  minuend; 
subtrahend;  remainder. — State  the  mental  acts 
involved  in  subtraction. — How  is  the  difference 
between  two  separate  numbers  found? — Describe 
the  sign  of  subtraction  and  state  its  use. — State 

125  and  illustrate  each  of  the  principles  of  subtrac- 
tion.— Under  what  condition  must  the  minuend  be 
prepared  before  a  subtraction  is  effected? — In  what 
consists  the  preparation? — Which  reduction  is  in- 
volved?— How  may  reduction  be  avoided  in  sub- 

130  traction? — Define    quotient;    division;    dividend; 

divisor. — Define  each  of  these  terms  in  the  light. of 

its  relation  to  multiplication. — What  is  meant  by  a 

.constant  subtraction? — State  how  division  grows 

out   of  subtraction? — How  is  division   related  to 


188 

135  multiplication? — In  what  consists  the  mental  act 
of  division? — Which  reduction  is  sometimes  in- 
volved in  division? — Illustrate. — State  and  illus- 
trate each  of  the  principles  of  division. — State  and 
illustrate  each  of  the  six  general  principles  of  divi- 

140  sion. — Define  disposition. — How  is  disposition  re- 
lated to  composition? — How  are  the  factors  of  a 
number  found? — State  and  illustrate  each  of  the 
principles  of  disposition. — Under  what  condition 
is  a  number  known  to  be  prime? — Define  evolu- 

145  tion ;  the  index  of  a  root. — Describe  and  state  the 
use  of  the  radical  sign  ;  exponent. — What  is  signi- 
fied by  a  fractional  exponent  other  than  a  frac- 
tional unit? — Under  how  many  and  what  heads  is 
the  analysis  of  numbers  discussed? — State  the  sub- 

150  stance  of  the  last  four  general  remarks  given  under 
analysis. — Define  g.  c.  d. — State  and  illustrate  the 
principles  under  this  head. — How  many  methods 
are  given  for  finding  g.  c.  d.? — Solve  an  example 
illustrative  of  each  method. — Define  1.  c.  m. — State 

155  the  principle  and  solve  an  example,  using  the  form 
given  on  page  53. — If  given  numbers  are  not 
readily  factored  how  may  their  1.  c.  m.  be  found? 
—Why? — Define  a  fraction,  giving  two  definitions. 
— According  to  the  primary  idea  of  a  fraction,  what 

160  is  the  maximum  value  of  a  fraction!  ? — How  is  an 
expression  like  1  to  be  interpreted? — How  is  a 
fraction  notated?— On  what  basis  are  fractions 
classified  as  decimal  and  common? — Define  a  deci- 
mal fraction. — How  is  a  decimal  fraction  usually 

165  notated? — How  may  it  be  notated? — What  is  a 
complex  decimal? — Define  a  common  fraction. — On 
what  basis  are  fractions  classified  as  proper  and 
improper? — Define  a  proper  fraction ;  an  improper 


189 

X 

fraction. — State  wherein  there  is  no  ground  for 

170  this  classification. — On  what  basis  are  fractions 
classified  as  simple,  compound  and  complex? — De- 
fine a  simple  fraction ;  a  compound  fraction  ;  a 
complex  fraction. — Show  wherein  this  classification 
is  not  well  founded. — State  the  likeness  of  a  frac- 

175  tion  to  division. — State  and  illustrate  each  of  the 
general  principles  of  fractions. — State  the  principle 
under  which  reduction  descending  of  fractions  may 
be  effected. — Under  what  principle  may  reduction 
ascending  of  fractions  be  effected? — How  may  a 

180  fraction  be  reduced  to  its  lowest  terms? — Under 
what  condition  is  a  common  fraction  not  reducible 
to  a  decimal? — Under  what  principle  is  addition  of 
fractions  effected? — Under  what  principle  is  sub- 
traction of  fractions  effected? — State  the  two  cases 

185  under  which  multiplication  of  fractions  is  present- 
ed.— Solve  an  example  under  each  case.— State  the 
two  cases  under  which  division  of  fractions  is  pre- 
sented — Solve  an  example  under  each  case,  using 
— a.,  common  fractions;  6.,  decimal  fractions.— 

190  Show  that  division  of  fractions  is  the  reverse  of 
multiplication  of  fractions. — Divide  -^  by  ^  effect- 
ing the  division  by  constant  subtraction. — Into 
what  classes  are  compound  numbers  divided? — On 
what  basis? — Write  the  diagram. — State  the  order 

195  to  be  pursued  in  the  study  of  a  "  measure." — Recite 
the  tables  of  compound  numbers. — How  many  and 
what  cases  of  reduction  descending  in  compound 
numbers? — Solve  an  example  under  each  case. 
How  many  and  what  cases  of  reduction  ascending 

200  in  compound  numbers? — Solve  an  example  under 
each  case. — Define  Longitude. — What  meridian  is 
generally  taken  as  prime? — Make  a  table  of  corres- 


190 

ponding  time  and  longitude  units. — What  is  meant 
by  relative  time? — Absolute  time? — Show  that  all 

205  places  have  not  the  same  relative  time. — Explain 
what  is  meant  by  "Standard  time," — Solve  exam- 
ples in  time  and  longitude. — What  is  area? — What 
is  a  volume,  or  solid? — Solve  an  example  in  which 
area  is  computed. — Solve  an  example  in  which 

210  volume  is  computed. — Show  that  length  and  breadth 
are  not  multiplied  together. — If  they  could  be 
would  the  product  be  surface? — Why? — Name  the 
three  primary  units  of  the  decimal  system  of  meas- 
ures*.— Which  of  these  is  the  fundamental  unit  of 

215  the  system? — How  does  the  meter  compare  in  length 
with  the  yard? — Name  the  land  unit. — What  is  its 
area? — By  what  units  are  most  surfaces  measured? 
— What  is  the  wood  unit? — What  is  its  volume? 
By  what  units  are  most  volumes  measured? — What 

220  units  are  used  in  measuring  great  weights? — How 
does  each  of  these  compare  with  the  gram? — Name 
each  prefix  designating  a  secondary  unit. — Con- 
struct tables  for  the  measures  of  length,  surface, 
volume,  capacity,  and  weight,  in  the  decimal  sys- 

225  tern. — What  relations  exist  by  means  of  which 
capacity  and  weight  are  readily  found  from  exten- 
sion?— What  is  percentage? — Name  the  essential 
terms  of  percentage. — Define  each. — Define  amount 
and  difference,  and  state  which  of  the  essential 

230  terms  includes  them. — State  the  relations  of  per- 
centage.— a.  To  multiplication. — b.  To  factoring. 
— c.  To  fractions. — State  each  of  the  general  canes 
of  percentage, — Give  the  solution  of  each, — How 
many  and  what  elements  are  involved  in  the  first 

235  class  of  applications  of  percentage? — Name  the 
principal  applications  of  the  first  class. — What  ele- 


191 

ments  are  involved  in  the  second  class  of  applica- 
tions of  percentage  ? — Name  the  principal  applica- 
tions of  the  second  class. — Define  profit  and  loss; 

240  cost;  selling  price;  gain;  loss. — State  the  relation 
of  profit  and  loss  to  percentage  by  naming  the  cor- 
responding terms. — What  is  an  agent? — A  factor? 
— A  broker? — A  commission  merchant? — Define 
commission ;  brokerage. — State  the  relation  of 

245  commission  and  brokerage  to  percentage  by  nam- 
ing the  corresponding  terms. — Define  stock  as  used 
in  percentage ;  a  corporation  ;  a  firm  ;  a  company ; 
a  share ;  a  certificate  of  stock  ;  a  dividend ;  par 
value;  market  value;  premium;  discount. — How 

250  is  stock  quoted  ? — State  the  relation  of  stock  to 
percentage  by  naming  the  corresponding  terms. 
—Define  insurance;  fire  insurance  ;  marine  insur- 
ance ;  life  insurance  ;  policy  ;  face  of  policy  ;  policy- 
holder  ;  underwriter ;  premium. — State  the  relation 

255  of  insurance  to  percentage  by  naming  the  corres- 
ponding terms. — Define  a  tax ;  a  property  tax ;  a 
poll  tax ;  an  income  tax ;  an  excise  tax ;  an  asses- 
sor; his  duties;  an  invoice;  tare;  leakage;  break- 
age; gross  weight;  net  weight;  specific  duty; 

260  advalorem  duty. — Define  interest;  principal  ; 
amount;  rate;  legal  rate. — What  is  the  time  unit 
in  interest? — Define  simple  interest;  a  partial  pay- 
ment.— State  the  United  States  rule  for  computing 
interest  in  partial  payments;  the  merchants'  rule. 

265  — How  many  elements  are  involved  in  simple  in- 
terest ? — What  relations  are  sustained  among  the 
elements? — Define' compound  interest;  annual  in- 
teiest;  discount;  true  discount;  present  worth. — 
State  the  relation  of  true  discount  to  simple  inter- 

270  est  by  naming  the  corresponding  terms. — Define 


192 

bank  discount;  term  of  discount;  proceeds;  days 
of  grace;  face  of  a  note;  legal  maturity;  a  protest. 
—If  an  interest  bearing  note  be  discounted  in 
bank,  on  what  is  the  discount  computed  ? — State 

275  the  relation  of  bank  discount  to  simple  interest  by 
naming  the  corresponding  terms. — Define  commer- 
cial discount. — State  its  relation  to  percentage. — 
Define  exchange;  a  draft ;  a  sight  draft;  a  time 
draft;  the  face  of  a  draft ;  the  course  of  exchange. 

280  — Who  is  the  drawer  or  maker  of  a  draft? — The 
drawee  ?- — The  payee  ? — State  the  relation  of  ex- 
change to  simple  interest  by  naming  the  corres- 
ponding terms. — What  is  meant  by  the  equation 
or  average  of  payments ?— Show  the  relation  of 

285  equation  of  payments  to  simple  interest. — Define 
Eatio. — How  many  and  what  terms  are  used  in 
thinking  a  ratio  ? — Define  each, — How  is  a  ratio 
notated  ? — Define  a  simple  ratio  ;  a  compound  ra- 
tio; an  inverse  ratio. — State  each  of  the  six  princi- 

290  pies  of  ratio. — Define  proportion  ?  a  simple  pro- 
portion ;  a  compound  proportion. — How  is  a  pro- 
portion notated? — State  and  demonstrate  the  prin- 
ciple of  proportion.  — Under  what  condition 
is  a  number  divided  into  proportional  parts? 

295  State  the  principles. — If  two  sides  of  a  right-angled 
triangle  be  known,  how  find  the  third  side  ? — The 
areas  of  similar  surfaces  are  to  each  other  as  what  ? 
How  find  the  area  of  a  trapezoid  ? — How  find  the 
two  sides  ot  a  rectangle,  if  the  area  and  the  ratio 

300  of  the  sides  are  given  ? — How  find  the  three  di- 
mensions ot  a  rectangular  solid,  if  the  solid  con- 
tents and  the  ratio  of  the  sides  are  given  ? — How 
find  the  convex  surface  of  a  cylinder? — How  find 
the  volume  of  a  cylinder  ? 


YB   17416 


M306O22 


PA  10 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


